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PEP:465
Title:A dedicated infix operator for matrix multiplication
Version:56b554e51e5a
Last-Modified:2014-04-09 23:57:23 -0400 (Wed, 09 Apr 2014)
Author:Nathaniel J. Smith <njs at pobox.com>
Status:Final
Type:Standards Track
Content-Type:text/x-rst
Created:20-Feb-2014
Python-Version:3.5
Post-History:13-Mar-2014

Abstract

This PEP proposes a new binary operator to be used for matrix multiplication, called @. (Mnemonic: @ is * for mATrices.)

Specification

A new binary operator is added to the Python language, together with the corresponding in-place version:

Op Precedence/associativity Methods
@ Same as * __matmul__, __rmatmul__
@= n/a __imatmul__

No implementations of these methods are added to the builtin or standard library types. However, a number of projects have reached consensus on the recommended semantics for these operations; see Intended usage details below for details.

For details on how this operator will be implemented in CPython, see Implementation details.

Motivation

Executive summary

In numerical code, there are two important operations which compete for use of Python's * operator: elementwise multiplication, and matrix multiplication. In the nearly twenty years since the Numeric library was first proposed, there have been many attempts to resolve this tension [13]; none have been really satisfactory. Currently, most numerical Python code uses * for elementwise multiplication, and function/method syntax for matrix multiplication; however, this leads to ugly and unreadable code in common circumstances. The problem is bad enough that significant amounts of code continue to use the opposite convention (which has the virtue of producing ugly and unreadable code in different circumstances), and this API fragmentation across codebases then creates yet more problems. There does not seem to be any good solution to the problem of designing a numerical API within current Python syntax -- only a landscape of options that are bad in different ways. The minimal change to Python syntax which is sufficient to resolve these problems is the addition of a single new infix operator for matrix multiplication.

Matrix multiplication has a singular combination of features which distinguish it from other binary operations, which together provide a uniquely compelling case for the addition of a dedicated infix operator:

  • Just as for the existing numerical operators, there exists a vast body of prior art supporting the use of infix notation for matrix multiplication across all fields of mathematics, science, and engineering; @ harmoniously fills a hole in Python's existing operator system.
  • @ greatly clarifies real-world code.
  • @ provides a smoother onramp for less experienced users, who are particularly harmed by hard-to-read code and API fragmentation.
  • @ benefits a substantial and growing portion of the Python user community.
  • @ will be used frequently -- in fact, evidence suggests it may be used more frequently than // or the bitwise operators.
  • @ allows the Python numerical community to reduce fragmentation, and finally standardize on a single consensus duck type for all numerical array objects.

Background: What's wrong with the status quo?

When we crunch numbers on a computer, we usually have lots and lots of numbers to deal with. Trying to deal with them one at a time is cumbersome and slow -- especially when using an interpreted language. Instead, we want the ability to write down simple operations that apply to large collections of numbers all at once. The n-dimensional array is the basic object that all popular numeric computing environments use to make this possible. Python has several libraries that provide such arrays, with numpy being at present the most prominent.

When working with n-dimensional arrays, there are two different ways we might want to define multiplication. One is elementwise multiplication:

[[1, 2],     [[11, 12],     [[1 * 11, 2 * 12],
 [3, 4]]  x   [13, 14]]  =   [3 * 13, 4 * 14]]

and the other is matrix multiplication [19]:

[[1, 2],     [[11, 12],     [[1 * 11 + 2 * 13, 1 * 12 + 2 * 14],
 [3, 4]]  x   [13, 14]]  =   [3 * 11 + 4 * 13, 3 * 12 + 4 * 14]]

Elementwise multiplication is useful because it lets us easily and quickly perform many multiplications on a large collection of values, without writing a slow and cumbersome for loop. And this works as part of a very general schema: when using the array objects provided by numpy or other numerical libraries, all Python operators work elementwise on arrays of all dimensionalities. The result is that one can write functions using straightforward code like a * b + c / d, treating the variables as if they were simple values, but then immediately use this function to efficiently perform this calculation on large collections of values, while keeping them organized using whatever arbitrarily complex array layout works best for the problem at hand.

Matrix multiplication is more of a special case. It's only defined on 2d arrays (also known as "matrices"), and multiplication is the only operation that has an important "matrix" version -- "matrix addition" is the same as elementwise addition; there is no such thing as "matrix bitwise-or" or "matrix floordiv"; "matrix division" and "matrix to-the-power-of" can be defined but are not very useful, etc. However, matrix multiplication is still used very heavily across all numerical application areas; mathematically, it's one of the most fundamental operations there is.

Because Python syntax currently allows for only a single multiplication operator *, libraries providing array-like objects must decide: either use * for elementwise multiplication, or use * for matrix multiplication. And, unfortunately, it turns out that when doing general-purpose number crunching, both operations are used frequently, and there are major advantages to using infix rather than function call syntax in both cases. Thus it is not at all clear which convention is optimal, or even acceptable; often it varies on a case-by-case basis.

Nonetheless, network effects mean that it is very important that we pick just one convention. In numpy, for example, it is technically possible to switch between the conventions, because numpy provides two different types with different __mul__ methods. For numpy.ndarray objects, * performs elementwise multiplication, and matrix multiplication must use a function call (numpy.dot). For numpy.matrix objects, * performs matrix multiplication, and elementwise multiplication requires function syntax. Writing code using numpy.ndarray works fine. Writing code using numpy.matrix also works fine. But trouble begins as soon as we try to integrate these two pieces of code together. Code that expects an ndarray and gets a matrix, or vice-versa, may crash or return incorrect results. Keeping track of which functions expect which types as inputs, and return which types as outputs, and then converting back and forth all the time, is incredibly cumbersome and impossible to get right at any scale. Functions that defensively try to handle both types as input and DTRT, find themselves floundering into a swamp of isinstance and if statements.

PEP 238 split / into two operators: / and //. Imagine the chaos that would have resulted if it had instead split int into two types: classic_int, whose __div__ implemented floor division, and new_int, whose __div__ implemented true division. This, in a more limited way, is the situation that Python number-crunchers currently find themselves in.

In practice, the vast majority of projects have settled on the convention of using * for elementwise multiplication, and function call syntax for matrix multiplication (e.g., using numpy.ndarray instead of numpy.matrix). This reduces the problems caused by API fragmentation, but it doesn't eliminate them. The strong desire to use infix notation for matrix multiplication has caused a number of specialized array libraries to continue to use the opposing convention (e.g., scipy.sparse, pyoperators, pyviennacl) despite the problems this causes, and numpy.matrix itself still gets used in introductory programming courses, often appears in StackOverflow answers, and so forth. Well-written libraries thus must continue to be prepared to deal with both types of objects, and, of course, are also stuck using unpleasant funcall syntax for matrix multiplication. After nearly two decades of trying, the numerical community has still not found any way to resolve these problems within the constraints of current Python syntax (see Rejected alternatives to adding a new operator below).

This PEP proposes the minimum effective change to Python syntax that will allow us to drain this swamp. It splits * into two operators, just as was done for /: * for elementwise multiplication, and @ for matrix multiplication. (Why not the reverse? Because this way is compatible with the existing consensus, and because it gives us a consistent rule that all the built-in numeric operators also apply in an elementwise manner to arrays; the reverse convention would lead to more special cases.)

So that's why matrix multiplication doesn't and can't just use *. Now, in the the rest of this section, we'll explain why it nonetheless meets the high bar for adding a new operator.

Why should matrix multiplication be infix?

Right now, most numerical code in Python uses syntax like numpy.dot(a, b) or a.dot(b) to perform matrix multiplication. This obviously works, so why do people make such a fuss about it, even to the point of creating API fragmentation and compatibility swamps?

Matrix multiplication shares two features with ordinary arithmetic operations like addition and multiplication on numbers: (a) it is used very heavily in numerical programs -- often multiple times per line of code -- and (b) it has an ancient and universally adopted tradition of being written using infix syntax. This is because, for typical formulas, this notation is dramatically more readable than any function call syntax. Here's an example to demonstrate:

One of the most useful tools for testing a statistical hypothesis is the linear hypothesis test for OLS regression models. It doesn't really matter what all those words I just said mean; if we find ourselves having to implement this thing, what we'll do is look up some textbook or paper on it, and encounter many mathematical formulas that look like:

S=(Hβ-r)T(HVHT) -1(Hβ-r)

Here the various variables are all vectors or matrices (details for the curious: [5]).

Now we need to write code to perform this calculation. In current numpy, matrix multiplication can be performed using either the function or method call syntax. Neither provides a particularly readable translation of the formula:

import numpy as np
from numpy.linalg import inv, solve

# Using dot function:
S = np.dot((np.dot(H, beta) - r).T,
           np.dot(inv(np.dot(np.dot(H, V), H.T)), np.dot(H, beta) - r))

# Using dot method:
S = (H.dot(beta) - r).T.dot(inv(H.dot(V).dot(H.T))).dot(H.dot(beta) - r)

With the @ operator, the direct translation of the above formula becomes:

S = (H @ beta - r).T @ inv(H @ V @ H.T) @ (H @ beta - r)

Notice that there is now a transparent, 1-to-1 mapping between the symbols in the original formula and the code that implements it.

Of course, an experienced programmer will probably notice that this is not the best way to compute this expression. The repeated computation of Hβ-r should perhaps be factored out; and, expressions of the form dot(inv(A), B) should almost always be replaced by the more numerically stable solve(A, B). When using @, performing these two refactorings gives us:

# Version 1 (as above)
S = (H @ beta - r).T @ inv(H @ V @ H.T) @ (H @ beta - r)

# Version 2
trans_coef = H @ beta - r
S = trans_coef.T @ inv(H @ V @ H.T) @ trans_coef

# Version 3
S = trans_coef.T @ solve(H @ V @ H.T, trans_coef)

Notice that when comparing between each pair of steps, it's very easy to see exactly what was changed. If we apply the equivalent transformations to the code using the .dot method, then the changes are much harder to read out or verify for correctness:

# Version 1 (as above)
S = (H.dot(beta) - r).T.dot(inv(H.dot(V).dot(H.T))).dot(H.dot(beta) - r)

# Version 2
trans_coef = H.dot(beta) - r
S = trans_coef.T.dot(inv(H.dot(V).dot(H.T))).dot(trans_coef)

# Version 3
S = trans_coef.T.dot(solve(H.dot(V).dot(H.T)), trans_coef)

Readability counts! The statements using @ are shorter, contain more whitespace, can be directly and easily compared both to each other and to the textbook formula, and contain only meaningful parentheses. This last point is particularly important for readability: when using function-call syntax, the required parentheses on every operation create visual clutter that makes it very difficult to parse out the overall structure of the formula by eye, even for a relatively simple formula like this one. Eyes are terrible at parsing non-regular languages. I made and caught many errors while trying to write out the 'dot' formulas above. I know they still contain at least one error, maybe more. (Exercise: find it. Or them.) The @ examples, by contrast, are not only correct, they're obviously correct at a glance.

If we are even more sophisticated programmers, and writing code that we expect to be reused, then considerations of speed or numerical accuracy might lead us to prefer some particular order of evaluation. Because @ makes it possible to omit irrelevant parentheses, we can be certain that if we do write something like (H @ V) @ H.T, then our readers will know that the parentheses must have been added intentionally to accomplish some meaningful purpose. In the dot examples, it's impossible to know which nesting decisions are important, and which are arbitrary.

Infix @ dramatically improves matrix code usability at all stages of programmer interaction.

Transparent syntax is especially crucial for non-expert programmers

A large proportion of scientific code is written by people who are experts in their domain, but are not experts in programming. And there are many university courses run each year with titles like "Data analysis for social scientists" which assume no programming background, and teach some combination of mathematical techniques, introduction to programming, and the use of programming to implement these mathematical techniques, all within a 10-15 week period. These courses are more and more often being taught in Python rather than special-purpose languages like R or Matlab.

For these kinds of users, whose programming knowledge is fragile, the existence of a transparent mapping between formulas and code often means the difference between succeeding and failing to write that code at all. This is so important that such classes often use the numpy.matrix type which defines * to mean matrix multiplication, even though this type is buggy and heavily disrecommended by the rest of the numpy community for the fragmentation that it causes. This pedagogical use case is, in fact, the only reason numpy.matrix remains a supported part of numpy. Adding @ will benefit both beginning and advanced users with better syntax; and furthermore, it will allow both groups to standardize on the same notation from the start, providing a smoother on-ramp to expertise.

But isn't matrix multiplication a pretty niche requirement?

The world is full of continuous data, and computers are increasingly called upon to work with it in sophisticated ways. Arrays are the lingua franca of finance, machine learning, 3d graphics, computer vision, robotics, operations research, econometrics, meteorology, computational linguistics, recommendation systems, neuroscience, astronomy, bioinformatics (including genetics, cancer research, drug discovery, etc.), physics engines, quantum mechanics, geophysics, network analysis, and many other application areas. In most or all of these areas, Python is rapidly becoming a dominant player, in large part because of its ability to elegantly mix traditional discrete data structures (hash tables, strings, etc.) on an equal footing with modern numerical data types and algorithms.

We all live in our own little sub-communities, so some Python users may be surprised to realize the sheer extent to which Python is used for number crunching -- especially since much of this particular sub-community's activity occurs outside of traditional Python/FOSS channels. So, to give some rough idea of just how many numerical Python programmers are actually out there, here are two numbers: In 2013, there were 7 international conferences organized specifically on numerical Python [3] [4]. At PyCon 2014, ~20% of the tutorials appear to involve the use of matrices [6].

To quantify this further, we used Github's "search" function to look at what modules are actually imported across a wide range of real-world code (i.e., all the code on Github). We checked for imports of several popular stdlib modules, a variety of numerically oriented modules, and various other extremely high-profile modules like django and lxml (the latter of which is the #1 most downloaded package on PyPI). Starred lines indicate packages which export array- or matrix-like objects which will adopt @ if this PEP is approved:

Count of Python source files on Github matching given search terms
                 (as of 2014-04-10, ~21:00 UTC)
================ ==========  ===============  =======  ===========
module           "import X"  "from X import"    total  total/numpy
================ ==========  ===============  =======  ===========
sys                 2374638            63301  2437939         5.85
os                  1971515            37571  2009086         4.82
re                  1294651             8358  1303009         3.12
numpy ************** 337916 ********** 79065 * 416981 ******* 1.00
warnings             298195            73150   371345         0.89
subprocess           281290            63644   344934         0.83
django                62795           219302   282097         0.68
math                 200084            81903   281987         0.68
threading            212302            45423   257725         0.62
pickle+cPickle       215349            22672   238021         0.57
matplotlib           119054            27859   146913         0.35
sqlalchemy            29842            82850   112692         0.27
pylab *************** 36754 ********** 41063 ** 77817 ******* 0.19
scipy *************** 40829 ********** 28263 ** 69092 ******* 0.17
lxml                  19026            38061    57087         0.14
zlib                  40486             6623    47109         0.11
multiprocessing       25247            19850    45097         0.11
requests              30896              560    31456         0.08
jinja2                 8057            24047    32104         0.08
twisted               13858             6404    20262         0.05
gevent                11309             8529    19838         0.05
pandas ************** 14923 *********** 4005 ** 18928 ******* 0.05
sympy                  2779             9537    12316         0.03
theano *************** 3654 *********** 1828 *** 5482 ******* 0.01
================ ==========  ===============  =======  ===========

These numbers should be taken with several grains of salt (see footnote for discussion: [12]), but, to the extent they can be trusted, they suggest that numpy might be the single most-imported non-stdlib module in the entire Pythonverse; it's even more-imported than such stdlib stalwarts as subprocess, math, pickle, and threading. And numpy users represent only a subset of the broader numerical community that will benefit from the @ operator. Matrices may once have been a niche data type restricted to Fortran programs running in university labs and military clusters, but those days are long gone. Number crunching is a mainstream part of modern Python usage.

In addition, there is some precedence for adding an infix operator to handle a more-specialized arithmetic operation: the floor division operator //, like the bitwise operators, is very useful under certain circumstances when performing exact calculations on discrete values. But it seems likely that there are many Python programmers who have never had reason to use // (or, for that matter, the bitwise operators). @ is no more niche than //.

So @ is good for matrix formulas, but how common are those really?

We've seen that @ makes matrix formulas dramatically easier to work with for both experts and non-experts, that matrix formulas appear in many important applications, and that numerical libraries like numpy are used by a substantial proportion of Python's user base. But numerical libraries aren't just about matrix formulas, and being important doesn't necessarily mean taking up a lot of code: if matrix formulas only occured in one or two places in the average numerically-oriented project, then it still wouldn't be worth adding a new operator. So how common is matrix multiplication, really?

When the going gets tough, the tough get empirical. To get a rough estimate of how useful the @ operator will be, the table below shows the rate at which different Python operators are actually used in the stdlib, and also in two high-profile numerical packages -- the scikit-learn machine learning library, and the nipy neuroimaging library -- normalized by source lines of code (SLOC). Rows are sorted by the 'combined' column, which pools all three code bases together. The combined column is thus strongly weighted towards the stdlib, which is much larger than both projects put together (stdlib: 411575 SLOC, scikit-learn: 50924 SLOC, nipy: 37078 SLOC). [7]

The dot row (marked ******) counts how common matrix multiply operations are in each codebase.

====  ======  ============  ====  ========
  op  stdlib  scikit-learn  nipy  combined
====  ======  ============  ====  ========
   =    2969          5536  4932      3376 / 10,000 SLOC
   -     218           444   496       261
   +     224           201   348       231
  ==     177           248   334       196
   *     156           284   465       192
   %     121           114   107       119
  **      59           111   118        68
  !=      40            56    74        44
   /      18           121   183        41
   >      29            70   110        39
  +=      34            61    67        39
   <      32            62    76        38
  >=      19            17    17        18
  <=      18            27    12        18
 dot ***** 0 ********** 99 ** 74 ****** 16
   |      18             1     2        15
   &      14             0     6        12
  <<      10             1     1         8
  //       9             9     1         8
  -=       5            21    14         8
  *=       2            19    22         5
  /=       0            23    16         4
  >>       4             0     0         3
   ^       3             0     0         3
   ~       2             4     5         2
  |=       3             0     0         2
  &=       1             0     0         1
 //=       1             0     0         1
  ^=       1             0     0         0
 **=       0             2     0         0
  %=       0             0     0         0
 <<=       0             0     0         0
 >>=       0             0     0         0
====  ======  ============  ====  ========

These two numerical packages alone contain ~780 uses of matrix multiplication. Within these packages, matrix multiplication is used more heavily than most comparison operators (< != <= >=). Even when we dilute these counts by including the stdlib into our comparisons, matrix multiplication is still used more often in total than any of the bitwise operators, and 2x as often as //. This is true even though the stdlib, which contains a fair amount of integer arithmetic and no matrix operations, makes up more than 80% of the combined code base.

By coincidence, the numeric libraries make up approximately the same proportion of the 'combined' codebase as numeric tutorials make up of PyCon 2014's tutorial schedule, which suggests that the 'combined' column may not be wildly unrepresentative of new Python code in general. While it's impossible to know for certain, from this data it seems entirely possible that across all Python code currently being written, matrix multiplication is already used more often than // and the bitwise operations.

But isn't it weird to add an operator with no stdlib uses?

It's certainly unusual (though extended slicing existed for some time builtin types gained support for it, Ellipsis is still unused within the stdlib, etc.). But the important thing is whether a change will benefit users, not where the software is being downloaded from. It's clear from the above that @ will be used, and used heavily. And this PEP provides the critical piece that will allow the Python numerical community to finally reach consensus on a standard duck type for all array-like objects, which is a necessary precondition to ever adding a numerical array type to the stdlib.

Compatibility considerations

Currently, the only legal use of the @ token in Python code is at statement beginning in decorators. The new operators are both infix; the one place they can never occur is at statement beginning. Therefore, no existing code will be broken by the addition of these operators, and there is no possible parsing ambiguity between decorator-@ and the new operators.

Another important kind of compatibility is the mental cost paid by users to update their understanding of the Python language after this change, particularly for users who do not work with matrices and thus do not benefit. Here again, @ has minimal impact: even comprehensive tutorials and references will only need to add a sentence or two to fully document this PEP's changes for a non-numerical audience.

Intended usage details

This section is informative, rather than normative -- it documents the consensus of a number of libraries that provide array- or matrix-like objects on how @ will be implemented.

This section uses the numpy terminology for describing arbitrary multidimensional arrays of data, because it is a superset of all other commonly used models. In this model, the shape of any array is represented by a tuple of integers. Because matrices are two-dimensional, they have len(shape) == 2, while 1d vectors have len(shape) == 1, and scalars have shape == (), i.e., they are "0 dimensional". Any array contains prod(shape) total entries. Notice that prod(()) == 1 [20] (for the same reason that sum(()) == 0); scalars are just an ordinary kind of array, not a special case. Notice also that we distinguish between a single scalar value (shape == (), analogous to 1), a vector containing only a single entry (shape == (1,), analogous to [1]), a matrix containing only a single entry (shape == (1, 1), analogous to [[1]]), etc., so the dimensionality of any array is always well-defined. Other libraries with more restricted representations (e.g., those that support 2d arrays only) might implement only a subset of the functionality described here.

Semantics

The recommended semantics for @ for different inputs are:

  • 2d inputs are conventional matrices, and so the semantics are obvious: we apply conventional matrix multiplication. If we write arr(2, 3) to represent an arbitrary 2x3 array, then arr(2, 3) @ arr(3, 4) returns an array with shape (2, 4).

  • 1d vector inputs are promoted to 2d by prepending or appending a '1' to the shape, the operation is performed, and then the added dimension is removed from the output. The 1 is always added on the "outside" of the shape: prepended for left arguments, and appended for right arguments. The result is that matrix @ vector and vector @ matrix are both legal (assuming compatible shapes), and both return 1d vectors; vector @ vector returns a scalar. This is clearer with examples.

    • arr(2, 3) @ arr(3, 1) is a regular matrix product, and returns an array with shape (2, 1), i.e., a column vector.
    • arr(2, 3) @ arr(3) performs the same computation as the previous (i.e., treats the 1d vector as a matrix containing a single column, shape = (3, 1)), but returns the result with shape (2,), i.e., a 1d vector.
    • arr(1, 3) @ arr(3, 2) is a regular matrix product, and returns an array with shape (1, 2), i.e., a row vector.
    • arr(3) @ arr(3, 2) performs the same computation as the previous (i.e., treats the 1d vector as a matrix containing a single row, shape = (1, 3)), but returns the result with shape (2,), i.e., a 1d vector.
    • arr(1, 3) @ arr(3, 1) is a regular matrix product, and returns an array with shape (1, 1), i.e., a single value in matrix form.
    • arr(3) @ arr(3) performs the same computation as the previous, but returns the result with shape (), i.e., a single scalar value, not in matrix form. So this is the standard inner product on vectors.

    An infelicity of this definition for 1d vectors is that it makes @ non-associative in some cases ((Mat1 @ vec) @ Mat2 != Mat1 @ (vec @ Mat2)). But this seems to be a case where practicality beats purity: non-associativity only arises for strange expressions that would never be written in practice; if they are written anyway then there is a consistent rule for understanding what will happen (Mat1 @ vec @ Mat2 is parsed as (Mat1 @ vec) @ Mat2, just like a - b - c); and, not supporting 1d vectors would rule out many important use cases that do arise very commonly in practice. No-one wants to explain to new users why to solve the simplest linear system in the obvious way, they have to type (inv(A) @ b[:, np.newaxis]).flatten() instead of inv(A) @ b, or perform an ordinary least-squares regression by typing solve(X.T @ X, X @ y[:, np.newaxis]).flatten() instead of solve(X.T @ X, X @ y). No-one wants to type (a[np.newaxis, :] @ b[:, np.newaxis])[0, 0] instead of a @ b every time they compute an inner product, or (a[np.newaxis, :] @ Mat @ b[:, np.newaxis])[0, 0] for general quadratic forms instead of a @ Mat @ b. In addition, sage and sympy (see below) use these non-associative semantics with an infix matrix multiplication operator (they use *), and they report that they haven't experienced any problems caused by it.

  • For inputs with more than 2 dimensions, we treat the last two dimensions as being the dimensions of the matrices to multiply, and 'broadcast' across the other dimensions. This provides a convenient way to quickly compute many matrix products in a single operation. For example, arr(10, 2, 3) @ arr(10, 3, 4) performs 10 separate matrix multiplies, each of which multiplies a 2x3 and a 3x4 matrix to produce a 2x4 matrix, and then returns the 10 resulting matrices together in an array with shape (10, 2, 4). The intuition here is that we treat these 3d arrays of numbers as if they were 1d arrays of matrices, and then apply matrix multiplication in an elementwise manner, where now each 'element' is a whole matrix. Note that broadcasting is not limited to perfectly aligned arrays; in more complicated cases, it allows several simple but powerful tricks for controlling how arrays are aligned with each other; see [10] for details. (In particular, it turns out that when broadcasting is taken into account, the standard scalar * matrix product is a special case of the elementwise multiplication operator *.)

    If one operand is >2d, and another operand is 1d, then the above rules apply unchanged, with 1d->2d promotion performed before broadcasting. E.g., arr(10, 2, 3) @ arr(3) first promotes to arr(10, 2, 3) @ arr(3, 1), then broadcasts the right argument to create the aligned operation arr(10, 2, 3) @ arr(10, 3, 1), multiplies to get an array with shape (10, 2, 1), and finally removes the added dimension, returning an array with shape (10, 2). Similarly, arr(2) @ arr(10, 2, 3) produces an intermediate array with shape (10, 1, 3), and a final array with shape (10, 3).

  • 0d (scalar) inputs raise an error. Scalar * matrix multiplication is a mathematically and algorithmically distinct operation from matrix @ matrix multiplication, and is already covered by the elementwise * operator. Allowing scalar @ matrix would thus both require an unnecessary special case, and violate TOOWTDI.

Adoption

We group existing Python projects which provide array- or matrix-like types based on what API they currently use for elementwise and matrix multiplication.

Projects which currently use * for elementwise multiplication, and function/method calls for matrix multiplication:

The developers of the following projects have expressed an intention to implement @ on their array-like types using the above semantics:

  • numpy
  • pandas
  • blaze
  • theano

The following projects have been alerted to the existence of the PEP, but it's not yet known what they plan to do if it's accepted. We don't anticipate that they'll have any objections, though, since everything proposed here is consistent with how they already do things:

  • pycuda
  • panda3d

Projects which currently use * for matrix multiplication, and function/method calls for elementwise multiplication:

The following projects have expressed an intention, if this PEP is accepted, to migrate from their current API to the elementwise-*, matmul-@ convention (i.e., this is a list of projects whose API fragmentation will probably be eliminated if this PEP is accepted):

  • numpy (numpy.matrix)
  • scipy.sparse
  • pyoperators
  • pyviennacl

The following projects have been alerted to the existence of the PEP, but it's not known what they plan to do if it's accepted (i.e., this is a list of projects whose API fragmentation may or may not be eliminated if this PEP is accepted):

  • cvxopt

Projects which currently use * for matrix multiplication, and which don't really care about elementwise multiplication of matrices:

There are several projects which implement matrix types, but from a very different perspective than the numerical libraries discussed above. These projects focus on computational methods for analyzing matrices in the sense of abstract mathematical objects (i.e., linear maps over free modules over rings), rather than as big bags full of numbers that need crunching. And it turns out that from the abstract math point of view, there isn't much use for elementwise operations in the first place; as discussed in the Background section above, elementwise operations are motivated by the bag-of-numbers approach. So these projects don't encounter the basic problem that this PEP exists to address, making it mostly irrelevant to them; while they appear superficially similar to projects like numpy, they're actually doing something quite different. They use * for matrix multiplication (and for group actions, and so forth), and if this PEP is accepted, their expressed intention is to continue doing so, while perhaps adding @ as an alias. These projects include:

  • sympy
  • sage

Implementation details

New functions operator.matmul and operator.__matmul__ are added to the standard library, with the usual semantics.

A corresponding function PyObject* PyObject_MatrixMultiply(PyObject *o1, PyObject *o2) is added to the C API.

A new AST node is added named MatMult, along with a new token ATEQUAL and new bytecode opcodes BINARY_MATRIX_MULTIPLY and INPLACE_MATRIX_MULTIPLY.

Two new type slots are added; whether this is to PyNumberMethods or a new PyMatrixMethods struct remains to be determined.

Rationale for specification details

Choice of operator

Why @ instead of some other spelling? There isn't any consensus across other programming languages about how this operator should be named [11]; here we discuss the various options.

Restricting ourselves only to symbols present on US English keyboards, the punctuation characters that don't already have a meaning in Python expression context are: @, backtick, $, !, and ?. Of these options, @ is clearly the best; ! and ? are already heavily freighted with inapplicable meanings in the programming context, backtick has been banned from Python by BDFL pronouncement (see PEP 3099), and $ is uglier, even more dissimilar to * and , and has Perl/PHP baggage. $ is probably the second-best option of these, though.

Symbols which are not present on US English keyboards start at a significant disadvantage (having to spend 5 minutes at the beginning of every numeric Python tutorial just going over keyboard layouts is not a hassle anyone really wants). Plus, even if we somehow overcame the typing problem, it's not clear there are any that are actually better than @. Some options that have been suggested include:

  • U+00D7 MULTIPLICATION SIGN: A × B
  • U+22C5 DOT OPERATOR: A ⋅ B
  • U+2297 CIRCLED TIMES: A ⊗ B
  • U+00B0 DEGREE: A ° B

What we need, though, is an operator that means "matrix multiplication, as opposed to scalar/elementwise multiplication". There is no conventional symbol with this meaning in either programming or mathematics, where these operations are usually distinguished by context. (And U+2297 CIRCLED TIMES is actually used conventionally to mean exactly the wrong things: elementwise multiplication -- the "Hadamard product" -- or outer product, rather than matrix/inner product like our operator). @ at least has the virtue that it looks like a funny non-commutative operator; a naive user who knows maths but not programming couldn't look at A * B versus A × B, or A * B versus A ⋅ B, or A * B versus A ° B and guess which one is the usual multiplication, and which one is the special case.

Finally, there is the option of using multi-character tokens. Some options:

  • Matlab and Julia use a .* operator. Aside from being visually confusable with *, this would be a terrible choice for us because in Matlab and Julia, * means matrix multiplication and .* means elementwise multiplication, so using .* for matrix multiplication would make us exactly backwards from what Matlab and Julia users expect.
  • APL apparently used +.×, which by combining a multi-character token, confusing attribute-access-like . syntax, and a unicode character, ranks somewhere below U+2603 SNOWMAN on our candidate list. If we like the idea of combining addition and multiplication operators as being evocative of how matrix multiplication actually works, then something like +* could be used -- though this may be too easy to confuse with *+, which is just multiplication combined with the unary + operator.
  • PEP 211 suggested ~*. This has the downside that it sort of suggests that there is a unary * operator that is being combined with unary ~, but it could work.
  • R uses %*% for matrix multiplication. In R this forms part of a general extensible infix system in which all tokens of the form %foo% are user-defined binary operators. We could steal the token without stealing the system.
  • Some other plausible candidates that have been suggested: >< (= ascii drawing of the multiplication sign ×); the footnote operator [*] or |*| (but when used in context, the use of vertical grouping symbols tends to recreate the nested parentheses visual clutter that was noted as one of the major downsides of the function syntax we're trying to get away from); ^*.

So, it doesn't matter much, but @ seems as good or better than any of the alternatives:

  • It's a friendly character that Pythoneers are already used to typing in decorators, but the decorator usage and the math expression usage are sufficiently dissimilar that it would be hard to confuse them in practice.
  • It's widely accessible across keyboard layouts (and thanks to its use in email addresses, this is true even of weird keyboards like those in phones).
  • It's round like * and .
  • The mATrices mnemonic is cute.
  • The swirly shape is reminiscent of the simultaneous sweeps over rows and columns that define matrix multiplication
  • Its asymmetry is evocative of its non-commutative nature.
  • Whatever, we have to pick something.

Precedence and associativity

There was a long discussion [15] about whether @ should be right- or left-associative (or even something more exotic [18]). Almost all Python operators are left-associative, so following this convention would be the simplest approach, but there were two arguments that suggested matrix multiplication might be worth making right-associative as a special case:

First, matrix multiplication has a tight conceptual association with function application/composition, so many mathematically sophisticated users have an intuition that an expression like RSx proceeds from right-to-left, with first S transforming the vector x, and then R transforming the result. This isn't universally agreed (and not all number-crunchers are steeped in the pure-math conceptual framework that motivates this intuition [16]), but at the least this intuition is more common than for other operations like 234 which everyone reads as going from left-to-right.

Second, if expressions like Mat @ Mat @ vec appear often in code, then programs will run faster (and efficiency-minded programmers will be able to use fewer parentheses) if this is evaluated as Mat @ (Mat @ vec) then if it is evaluated like (Mat @ Mat) @ vec.

However, weighing against these arguments are the following:

Regarding the efficiency argument, empirically, we were unable to find any evidence that Mat @ Mat @ vec type expressions actually dominate in real-life code. Parsing a number of large projects that use numpy, we found that when forced by numpy's current funcall syntax to choose an order of operations for nested calls to dot, people actually use left-associative nesting slightly more often than right-associative nesting [17]. And anyway, writing parentheses isn't so bad -- if an efficiency-minded programmer is going to take the trouble to think through the best way to evaluate some expression, they probably should write down the parentheses regardless of whether they're needed, just to make it obvious to the next reader that they order of operations matter.

In addition, it turns out that other languages, including those with much more of a focus on linear algebra, overwhelmingly make their matmul operators left-associative. Specifically, the @ equivalent is left-associative in R, Matlab, Julia, IDL, and Gauss. The only exceptions we found are Mathematica, in which a @ b @ c would be parsed non-associatively as dot(a, b, c), and APL, in which all operators are right-associative. There do not seem to exist any languages that make @ right-associative and * left-associative. And these decisions don't seem to be controversial -- I've never seen anyone complaining about this particular aspect of any of these other languages, and the left-associativity of * doesn't seem to bother users of the existing Python libraries that use * for matrix multiplication. So, at the least we can conclude from this that making @ left-associative will certainly not cause any disasters. Making @ right-associative, OTOH, would be exploring new and uncertain ground.

And another advantage of left-associativity is that it is much easier to learn and remember that @ acts like *, than it is to remember first that @ is unlike other Python operators by being right-associative, and then on top of this, also have to remember whether it is more tightly or more loosely binding than *. (Right-associativity forces us to choose a precedence, and intuitions were about equally split on which precedence made more sense. So this suggests that no matter which choice we made, no-one would be able to guess or remember it.)

On net, therefore, the general consensus of the numerical community is that while matrix multiplication is something of a special case, it's not special enough to break the rules, and @ should parse like * does.

(Non)-Definitions for built-in types

No __matmul__ or __matpow__ are defined for builtin numeric types (float, int, etc.) or for the numbers.Number hierarchy, because these types represent scalars, and the consensus semantics for @ are that it should raise an error on scalars.

We do not -- for now -- define a __matmul__ method on the standard memoryview or array.array objects, for several reasons. Of course this could be added if someone wants it, but these types would require quite a bit of additional work beyond __matmul__ before they could be used for numeric work -- e.g., they have no way to do addition or scalar multiplication either! -- and adding such functionality is beyond the scope of this PEP. In addition, providing a quality implementation of matrix multiplication is highly non-trivial. Naive nested loop implementations are very slow and shipping such an implementation in CPython would just create a trap for users. But the alternative -- providing a modern, competitive matrix multiply -- would require that CPython link to a BLAS library, which brings a set of new complications. In particular, several popular BLAS libraries (including the one that ships by default on OS X) currently break the use of multiprocessing [8]. Together, these considerations mean that the cost/benefit of adding __matmul__ to these types just isn't there, so for now we'll continue to delegate these problems to numpy and friends, and defer a more systematic solution to a future proposal.

There are also non-numeric Python builtins which define __mul__ (str, list, ...). We do not define __matmul__ for these types either, because why would we even do that.

Non-definition of matrix power

Earlier versions of this PEP also proposed a matrix power operator, @@, analogous to **. But on further consideration, it was decided that the utility of this was sufficiently unclear that it would be better to leave it out for now, and only revisit the issue if -- once we have more experience with @ -- it turns out that @@ is truly missed. [14]

Rejected alternatives to adding a new operator

Over the past few decades, the Python numeric community has explored a variety of ways to resolve the tension between matrix and elementwise multiplication operations. PEP 211 and PEP 225, both proposed in 2000 and last seriously discussed in 2008 [9], were early attempts to add new operators to solve this problem, but suffered from serious flaws; in particular, at that time the Python numerical community had not yet reached consensus on the proper API for array objects, or on what operators might be needed or useful (e.g., PEP 225 proposes 6 new operators with unspecified semantics). Experience since then has now led to consensus that the best solution, for both numeric Python and core Python, is to add a single infix operator for matrix multiply (together with the other new operators this implies like @=).

We review some of the rejected alternatives here.

Use a second type that defines __mul__ as matrix multiplication: As discussed above (Background: What's wrong with the status quo?), this has been tried this for many years via the numpy.matrix type (and its predecessors in Numeric and numarray). The result is a strong consensus among both numpy developers and developers of downstream packages that numpy.matrix should essentially never be used, because of the problems caused by having conflicting duck types for arrays. (Of course one could then argue we should only define __mul__ to be matrix multiplication, but then we'd have the same problem with elementwise multiplication.) There have been several pushes to remove numpy.matrix entirely; the only counter-arguments have come from educators who find that its problems are outweighed by the need to provide a simple and clear mapping between mathematical notation and code for novices (see Transparent syntax is especially crucial for non-expert programmers). But, of course, starting out newbies with a dispreferred syntax and then expecting them to transition later causes its own problems. The two-type solution is worse than the disease.

Add lots of new operators, or add a new generic syntax for defining infix operators: In addition to being generally un-Pythonic and repeatedly rejected by BDFL fiat, this would be using a sledgehammer to smash a fly. The scientific python community has consensus that adding one operator for matrix multiplication is enough to fix the one otherwise unfixable pain point. (In retrospect, we all think PEP 225 was a bad idea too -- or at least far more complex than it needed to be.)

Add a new @ (or whatever) operator that has some other meaning in general Python, and then overload it in numeric code: This was the approach taken by PEP 211, which proposed defining @ to be the equivalent of itertools.product. The problem with this is that when taken on its own terms, it's pretty clear that itertools.product doesn't actually need a dedicated operator. It hasn't even been deemed worth of a builtin. (During discussions of this PEP, a similar suggestion was made to define @ as a general purpose function composition operator, and this suffers from the same problem; functools.compose isn't even useful enough to exist.) Matrix multiplication has a uniquely strong rationale for inclusion as an infix operator. There almost certainly don't exist any other binary operations that will ever justify adding any other infix operators to Python.

Add a .dot method to array types so as to allow "pseudo-infix" A.dot(B) syntax: This has been in numpy for some years, and in many cases it's better than dot(A, B). But it's still much less readable than real infix notation, and in particular still suffers from an extreme overabundance of parentheses. See Why should matrix multiplication be infix? above.

Use a 'with' block to toggle the meaning of * within a single code block: E.g., numpy could define a special context object so that we'd have:

c = a * b   # element-wise multiplication
with numpy.mul_as_dot:
    c = a * b  # matrix multiplication

However, this has two serious problems: first, it requires that every array-like type's __mul__ method know how to check some global state (numpy.mul_is_currently_dot or whatever). This is fine if a and b are numpy objects, but the world contains many non-numpy array-like objects. So this either requires non-local coupling -- every numpy competitor library has to import numpy and then check numpy.mul_is_currently_dot on every operation -- or else it breaks duck-typing, with the above code doing radically different things depending on whether a and b are numpy objects or some other sort of object. Second, and worse, with blocks are dynamically scoped, not lexically scoped; i.e., any function that gets called inside the with block will suddenly find itself executing inside the mul_as_dot world, and crash and burn horribly -- if you're lucky. So this is a construct that could only be used safely in rather limited cases (no function calls), and which would make it very easy to shoot yourself in the foot without warning.

Use a language preprocessor that adds extra numerically-oriented operators and perhaps other syntax: (As per recent BDFL suggestion: [1]) This suggestion seems based on the idea that numerical code needs a wide variety of syntax additions. In fact, given @, most numerical users don't need any other operators or syntax; it solves the one really painful problem that cannot be solved by other means, and that causes painful reverberations through the larger ecosystem. Defining a new language (presumably with its own parser which would have to be kept in sync with Python's, etc.), just to support a single binary operator, is neither practical nor desireable. In the numerical context, Python's competition is special-purpose numerical languages (Matlab, R, IDL, etc.). Compared to these, Python's killer feature is exactly that one can mix specialized numerical code with code for XML parsing, web page generation, database access, network programming, GUI libraries, and so forth, and we also gain major benefits from the huge variety of tutorials, reference material, introductory classes, etc., which use Python. Fragmenting "numerical Python" from "real Python" would be a major source of confusion. A major motivation for this PEP is to reduce fragmentation. Having to set up a preprocessor would be an especially prohibitive complication for unsophisticated users. And we use Python because we like Python! We don't want almost-but-not-quite-Python.

Use overloading hacks to define a "new infix operator" like *dot*, as in a well-known Python recipe: (See: [2]) Beautiful is better than ugly. This is... not beautiful. And not Pythonic. And especially unfriendly to beginners, who are just trying to wrap their heads around the idea that there's a coherent underlying system behind these magic incantations that they're learning, when along comes an evil hack like this that violates that system, creates bizarre error messages when accidentally misused, and whose underlying mechanisms can't be understood without deep knowledge of how object oriented systems work.

Use a special "facade" type to support syntax like arr.M * arr: This is very similar to the previous proposal, in that the .M attribute would basically return the same object as arr *dot` would, and thus suffers the same objections about 'magicalness'.  This approach also has some non-obvious complexities: for example, while ``arr.M * arr must return an array, arr.M * arr.M and arr * arr.M must return facade objects, or else arr.M * arr.M * arr and arr * arr.M * arr will not work. But this means that facade objects must be able to recognize both other array objects and other facade objects (which creates additional complexity for writing interoperating array types from different libraries who must now recognize both each other's array types and their facade types). It also creates pitfalls for users who may easily type arr * arr.M or arr.M * arr.M and expect to get back an array object; instead, they will get a mysterious object that throws errors when they attempt to use it. Basically with this approach users must be careful to think of .M* as an indivisible unit that acts as an infix operator -- and as infix-operator-like token strings go, at least *dot* is prettier looking (look at its cute little ears!).

Discussions of this PEP

Collected here for reference:

References

[1]From a comment by GvR on a G+ post by GvR; the comment itself does not seem to be directly linkable: https://plus.google.com/115212051037621986145/posts/hZVVtJ9bK3u
[2]http://code.activestate.com/recipes/384122-infix-operators/ http://www.sagemath.org/doc/reference/misc/sage/misc/decorators.html#sage.misc.decorators.infix_operator
[3]http://conference.scipy.org/past.html
[4]http://pydata.org/events/
[5]

In this formula, β is a vector or matrix of regression coefficients, V is the estimated variance/covariance matrix for these coefficients, and we want to test the null hypothesis that Hβ=r; a large S then indicates that this hypothesis is unlikely to be true. For example, in an analysis of human height, the vector β might contain one value which was the the average height of the measured men, and another value which was the average height of the measured women, and then setting H=[1,-1],r=0 would let us test whether men and women are the same height on average. Compare to eq. 2.139 in http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/tutorials/xegbohtmlnode17.html

Example code is adapted from https://github.com/rerpy/rerpy/blob/0d274f85e14c3b1625acb22aed1efa85d122ecb7/rerpy/incremental_ls.py#L202

[6]

Out of the 36 tutorials scheduled for PyCon 2014 (https://us.pycon.org/2014/schedule/tutorials/), we guess that the 8 below will almost certainly deal with matrices:

  • Dynamics and control with Python
  • Exploring machine learning with Scikit-learn
  • How to formulate a (science) problem and analyze it using Python code
  • Diving deeper into Machine Learning with Scikit-learn
  • Data Wrangling for Kaggle Data Science Competitions – An etude
  • Hands-on with Pydata: how to build a minimal recommendation engine.
  • Python for Social Scientists
  • Bayesian statistics made simple

In addition, the following tutorials could easily involve matrices:

  • Introduction to game programming
  • mrjob: Snakes on a Hadoop ("We'll introduce some data science concepts, such as user-user similarity, and show how to calculate these metrics...")
  • Mining Social Web APIs with IPython Notebook
  • Beyond Defaults: Creating Polished Visualizations Using Matplotlib

This gives an estimated range of 8 to 12 / 36 = 22% to 33% of tutorials dealing with matrices; saying ~20% then gives us some wiggle room in case our estimates are high.

[7]

SLOCs were defined as physical lines which contain at least one token that is not a COMMENT, NEWLINE, ENCODING, INDENT, or DEDENT. Counts were made by using tokenize module from Python 3.2.3 to examine the tokens in all files ending .py underneath some directory. Only tokens which occur at least once in the source trees are included in the table. The counting script is available in the PEP repository.

Matrix multiply counts were estimated by counting how often certain tokens which are used as matrix multiply function names occurred in each package. This creates a small number of false positives for scikit-learn, because we also count instances of the wrappers around dot that this package uses, and so there are a few dozen tokens which actually occur in import or def statements.

All counts were made using the latest development version of each project as of 21 Feb 2014.

'stdlib' is the contents of the Lib/ directory in commit d6aa3fa646e2 to the cpython hg repository, and treats the following tokens as indicating matrix multiply: n/a.

'scikit-learn' is the contents of the sklearn/ directory in commit 69b71623273ccfc1181ea83d8fb9e05ae96f57c7 to the scikit-learn repository (https://github.com/scikit-learn/scikit-learn), and treats the following tokens as indicating matrix multiply: dot, fast_dot, safe_sparse_dot.

'nipy' is the contents of the nipy/ directory in commit 5419911e99546401b5a13bd8ccc3ad97f0d31037 to the nipy repository (https://github.com/nipy/nipy/), and treats the following tokens as indicating matrix multiply: dot.

[8]BLAS libraries have a habit of secretly spawning threads, even when used from single-threaded programs. And threads play very poorly with fork(); the usual symptom is that attempting to perform linear algebra in a child process causes an immediate deadlock.
[9]http://fperez.org/py4science/numpy-pep225/numpy-pep225.html
[10]http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html
[11]http://mail.scipy.org/pipermail/scipy-user/2014-February/035499.html
[12]

Counts were produced by manually entering the string "import foo" or "from foo import" (with quotes) into the Github code search page, e.g.: https://github.com/search?q=%22import+numpy%22&ref=simplesearch&type=Code on 2014-04-10 at ~21:00 UTC. The reported values are the numbers given in the "Languages" box on the lower-left corner, next to "Python". This also causes some undercounting (e.g., leaving out Cython code, and possibly one should also count HTML docs and so forth), but these effects are negligible (e.g., only ~1% of numpy usage appears to occur in Cython code, and probably even less for the other modules listed). The use of this box is crucial, however, because these counts appear to be stable, while the "overall" counts listed at the top of the page ("We've found ___ code results") are highly variable even for a single search -- simply reloading the page can cause this number to vary by a factor of 2 (!!). (They do seem to settle down if one reloads the page repeatedly, but nonetheless this is spooky enough that it seemed better to avoid these numbers.)

These numbers should of course be taken with multiple grains of salt; it's not clear how representative Github is of Python code in general, and limitations of the search tool make it impossible to get precise counts. AFAIK this is the best data set currently available, but it'd be nice if it were better. In particular:

  • Lines like import sys, os will only be counted in the sys row.
  • A file containing both import X and from X import will be counted twice
  • Imports of the form from X.foo import ... are missed. We could catch these by instead searching for "from X", but this is a common phrase in English prose, so we'd end up with false positives from comments, strings, etc. For many of the modules considered this shouldn't matter too much -- for example, the stdlib modules have flat namespaces -- but it might especially lead to undercounting of django, scipy, and twisted.

Also, it's possible there exist other non-stdlib modules we didn't think to test that are even more-imported than numpy -- though we tried quite a few of the obvious suspects. If you find one, let us know! The modules tested here were chosen based on a combination of intuition and the top-100 list at pypi-ranking.info.

Fortunately, it doesn't really matter if it turns out that numpy is, say, merely the third most-imported non-stdlib module, since the point is just that numeric programming is a common and mainstream activity.

Finally, we should point out the obvious: whether a package is import**ed** is rather different from whether it's import**ant**. No-one's claiming numpy is "the most important package" or anything like that. Certainly more packages depend on distutils, e.g., then depend on numpy -- and far fewer source files import distutils than import numpy. But this is fine for our present purposes. Most source files don't import distutils because most source files don't care how they're distributed, so long as they are; these source files thus don't care about details of how distutils' API works. This PEP is in some sense about changing how numpy's and related packages' APIs work, so the relevant metric is to look at source files that are choosing to directly interact with that API, which is sort of like what we get by looking at import statements.

[13]The first such proposal occurs in Jim Hugunin's very first email to the matrix SIG in 1995, which lays out the first draft of what became Numeric. He suggests using * for elementwise multiplication, and % for matrix multiplication: https://mail.python.org/pipermail/matrix-sig/1995-August/000002.html
[14]http://mail.scipy.org/pipermail/numpy-discussion/2014-March/069502.html
[15]http://mail.scipy.org/pipermail/numpy-discussion/2014-March/069444.html http://mail.scipy.org/pipermail/numpy-discussion/2014-March/069605.html
[16]http://mail.scipy.org/pipermail/numpy-discussion/2014-March/069610.html
[17]http://mail.scipy.org/pipermail/numpy-discussion/2014-March/069578.html
[18]http://mail.scipy.org/pipermail/numpy-discussion/2014-March/069530.html
[19]https://en.wikipedia.org/wiki/Matrix_multiplication
[20]https://en.wikipedia.org/wiki/Empty_product