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In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic. On the other hand, matrices are arrays of numbers, so there is no unique way to define "the" multiplication of matrices. As such, in general the term "matrix multiplication" refers to a number of different ways to multiply matrices. The key features of any matrix multiplication include: the number of rows and columns the original matrices have (called the "size", "order" or "dimension"), and specifying how the entries of the matrices generate the new matrix.
Like vectors, matrices of any size can be multiplied by scalars, which amounts to multiplying every entry of the matrix by the same number. Similar to the entrywise definition of adding or subtracting matrices, multiplication of two matrices of the same size can be defined by multiplying the corresponding entries, and this is known as the Hadamard product.
One can form many other definitions. However, the most useful definition can be motivated by linear equations and linear transformations on vectors, which have numerous applications in applied mathematics, physics, and engineering. This definition is often called the matrix product.^{[1]}^{[2]} Although this definition is not commutative, it still retains the associative property and is distributive over entrywise addition of matrices, and leads to the definitions of: an identity matrix (analogous to multiplying real numbers by the number 1), an inverse matrix (analogous to the multiplicative inverse of a number), powers and nth roots of a square matrix, consequently the matrix exponential can be defined by a power series of a square matrix, and so on. A consequence of this matrix product is determinant multiplicativity. This matrix product is an important operation in matrix groups, and the theory of group representations and irreps.
This article will use the following notational conventions: matrices are represented by capital letters in bold, vectors in lowercase bold, and entries of vectors and matrices are italic (since they are scalars). Index notation is often the clearest way to express definitions, and will be used as standard in the literature.
The simplest form of multiplication associated with matrices is scalar multiplication.
The left multiplication of a matrix A with a scalar λ gives another matrix λA of the same size as A. The entries of λA are given by
explicitly:
Similarly, the right multiplication of a matrix A with a scalar λ is defined to be
When the underlying ring is commutative, for example, the real or complex number field, these two multiplications are the same, and are simply called scalar multiplication. However, for matrices over a more general ring that are not commutative, such as the quaternions, they may not be equal.
For a real scalar and matrix:
For quaternion scalars and matrices:
Assume two matrices are to be multiplied (the generalization to any number is discussed below). If A is an n×m matrix and B is an m×p matrix, the result AB of their multiplication is an n×p matrix defined only if the number of columns m in A is equal to the number of rows m in B.
When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix (depicted in the image right). One may compute each entry in the third matrix one at a time.
For two matrices
where necessarily the number of columns in A equals the number of rows in B, in this case m, the matrix product AB is denoted without symbol (no multiplication signs or dots) to be the n×p matrix:^{[3]}^{[4]}
where AB has entries defined by:
Treating the rows and columns in each matrix as row and column vectors respectively, this entry is also their vector dot product:
(See below for further details). Usually the entries are numbers or expressions, but can even be matrices themselves (see block matrix). The matrix product can still be calculated exactly the same way.
The figure to the right illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the product matrix corresponds to a row of A and a column of B.
The values at the intersections marked with circles are:
If
their matrix product is:
yet BA is not defined.
The product of a square matrix multiplied by a column matrix arises naturally in linear algebra; for solving linear equations and representing linear transformations. By choosing a, b, c, d in A appropriately, A can represent a variety of transformations such as rotations, scaling and reflections, shears, of a geometric shape in space.
If
their matrix products are:
and
In this case, both products AB and BA are defined, and the entries show that AB and BA are not equal in general. Multiplying square matrices which represent linear transformations corresponds to the composite transformation (see below for details). Also, similarity transformations involving similar matrices are matrix products of square matrices, which requires the notion of an inverse matrix (see also below).
Analogous to numbers (elements of a field), matrices satisfy the following general properties, although there is one subtlety, due to the nature of matrix multiplication.
Matrices offer a concise way of representing linear transformations between vector spaces, and matrix multiplication corresponds to the composition of linear transformations. The matrix product of two matrices can be defined when their entries belong to the same ring, and hence can be added and multiplied.
Let U, V, and W be vector spaces over the same field with given bases, S: V → W and T: U → V be linear transformations and ST: U → W be their composition.
Suppose that A, B, and C are the matrices representing the transformations S, T, and ST with respect to the given bases.
Then AB = C, that is, the matrix of the composition (or the product) of linear transformations is the product of their matrices with respect to the given bases.
Matrix multiplication can be extended to the case of more than two matrices, provided that for each sequential pair, their dimensions match.
The product of N matrices A_{1}, A_{2}, ..., A_{N} with sizes n_{0}×n_{1}, n_{1}×n_{2}, ..., n_{N − 1}×n_{N}, is the n_{0}×n_{N} matrix:
The same properties will hold, as long as the ordering of matrices is not changed. The number of possible ways of grouping n matrices for multiplication is equal to the (n − 1)th Catalan number.
For example, if A, B, C, and D are respectively m×p, p×q, q×r, and r×n matrices, then there are 5 ways of grouping them without changing their order, and
is an m×n matrix.
Square matrices can be multiplied by themselves repeatedly in the same way as ordinary numbers, because they always have the same number of rows and columns. This repeated multiplication can be described as a power of the matrix, a special case of the ordinary matrix product. On the contrary, rectangular matrices do not have the same number of rows and columns so they can never be raised to a power. An n×n matrix A raised to a positive integer k is defined as
and the following identities hold, where λ is a scalar:
where I is the identity matrix. This is parallel to the zeroth power of any number which equals unity.
The naive computation of matrix powers is to multiply k times the matrix A to the result, starting with the identity matrix just like the scalar case. This can be improved using exponentiation by squaring, a method commonly used for scalars. For diagonalizable matrices, an even better method is to use the eigenvalue decomposition of A. Another method based on the Cayley–Hamilton theorem finds an identity using the matrices' characteristic polynomial, producing a more effective equation for A^{k} in which a scalar is raised to the required power, rather than an entire matrix.
A special case is the power of a diagonal matrix A.
Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the power k of a diagonal matrix A will have entries raised to the power. Explicitly;
meaning it is easy to raise a diagonal matrix to a power. When raising an arbitrary matrix (not necessarily a diagonal matrix) to a power, it is often helpful to exploit this property by diagonalizing the matrix first.
Given two column vectors a and b, the Euclidean inner product and outer product are the simplest special cases of the matrix product, by transposing the column vectors into row vectors.^{[5]}
is a column vector multiplied on the left by a row vector:
More explicitly,
is a row vector multiplied on the left by a column vector:
where
Suppose that the first n×m matrix A is decomposed into its row vectors a_{i}, and the second m×p matrix B into its column vectors b_{i}:^{[1]}
where
The entries in the introduction were given by:
It is also possible to express a matrix product in terms of concatenations of products of matrices and row or column vectors:
These decompositions are particularly useful for matrices that are envisioned as concatenations of particular types of row vectors or column vectors, e.g. orthogonal matrices (whose rows and columns are unit vectors orthogonal to each other) and Markov matrices (whose rows or columns sum to 1).
An alternative method results when the decomposition is done the other way around, i.e. the first matrix A is decomposed into column vectors a_{i} and the second matrix B into row vectors b_{i}:
where this time
This method emphasizes the effect of individual column/row pairs on the result, which is a useful point of view with e.g. covariance matrices, where each such pair corresponds to the effect of a single sample point.
What is the fastest algorithm for matrix multiplication? 
The running time of square matrix multiplication, if carried out naïvely, is . The running time for multiplying rectangular matrices (one m×pmatrix with one p×nmatrix) is , however, more efficient algorithms exist, such as Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". It is based on a way of multiplying two 2×2matrices which requires only 7 multiplications (instead of the usual 8), at the expense of several additional addition and subtraction operations. Applying this recursively gives an algorithm with a multiplicative cost of . Strassen's algorithm is more complex, and the numerical stability is reduced compared to the naïve algorithm.^{[6]} Nevertheless, it appears in several libraries, such as BLAS, where it is significantly more efficient for matrices with dimensions n > 100,^{[7]} and is very useful for large matrices over exact domains such as finite fields, where numerical stability is not an issue.
The current algorithm with the lowest known exponent k is a generalization of the Coppersmith–Winograd algorithm that has an asymptotic complexity of O(n^{2.3727}) thanks to Vassilevska Williams.^{[8]} This algorithm, and the CoppersmithWinograd algorithm on which it is based, are similar to Strassen's algorithm: a way is devised for multiplying two k×kmatrices with fewer than k^{3} multiplications, and this technique is applied recursively. However, the constant coefficient hidden by the Big O notation is so large that these algorithms are only worthwhile for matrices that are too large to handle on presentday computers.^{[9]}
Since any algorithm for multiplying two n×nmatrices has to process all 2×n^{2}entries, there is an asymptotic lower bound of operations. Raz (2002) proves a lower bound of for bounded coefficient arithmetic circuits over the real or complex numbers.
Cohn et al. (2003, 2005) put methods such as the Strassen and Coppersmith–Winograd algorithms in an entirely different grouptheoretic context, by utilising triples of subsets of finite groups which satisfy a disjointness property called the triple product property (TPP). They show that if families of wreath products of Abelian groups with symmetric groups realise families of subset triples with a simultaneous version of the TPP, then there are matrix multiplication algorithms with essentially quadratic complexity. Most researchers believe that this is indeed the case.^{[10]} However, Alon, Shpilka and Umans have recently shown that some of these conjectures implying fast matrix multiplication are incompatible with another plausible conjecture, the sunflower conjecture.^{[11]}
Because of the nature of matrix operations and the layout of matrices in memory, it is typically possible to gain substantial performance gains through use of parallelization and vectorization. It should therefore be noted that some lower timecomplexity algorithms on paper may have indirect time complexity costs on real machines.
Freivalds' algorithm is a simple Monte Carlo algorithm that given matrices verifies in time if .
On modern architectures with hierarchical memory, the cost of loading and storing input matrix elements tends to dominate the cost of arithmetic. On a single machine this is the amount of data transferred between RAM and cache, while on a distributed memory multinode machine it is the amount transferred between nodes; in either case it is called the communication bandwidth. The naïve algorithm using three nested loops uses communication bandwidth.
Cannon's algorithm, also known as the 2D algorithm, partitions each input matrix into a block matrix whose elements are submatrices of size by , where M is the size of fast memory.^{[12]} The naïve algorithm is then used over the block matrices, computing products of submatrices entirely in fast memory. This reduces communication bandwidth to , which is asymptotically optimal (for algorithms performing computation).^{[13]}^{[14]}
In a distributed setting with p processors arranged in a by 2D mesh, one submatrix of the result can be assigned to each processor, and the product can be computed with each processor transmitting words, which is asymptotically optimal assuming that each node stores the minimum elements.^{[14]} This can be improved by the 3D algorithm, which arranges the processors in a 3D cube mesh, assigning every product of two input submatrices to a single processor. The result submatrices are then generated by performing a reduction over each row.^{[15]} This algorithm transmits words per processor, which is asymptotically optimal.^{[14]} However, this requires replicating each input matrix element times, and so requires a factor of more memory than is needed to store the inputs. This algorithm can be combined with Strassen to further reduce runtime.^{[15]} "2.5D" algorithms provide a continuous tradeoff between memory usage and communication bandwidth.^{[16]}
There are other ways to multiply two matrices, in fact simpler than the definition above.
For two matrices of the same dimensions, there is the Hadamard product, also known as the elementwise product, pointwise product, entrywise product and the Schur product.^{[17]} For two matrices A and B of the same dimensions, the Hadamard product A ○ B is a matrix of the same dimensions, which has elements
explicitly:
Due to the characteristic entrywise procedure, this operation is identical to many multiplying ordinary numbers (mn of them) all at once; hence the Hadamard product is commutative, associative and distributive over addition, and is a principal submatrix of the Kronecker product. It appears in lossy compression algorithms such as JPEG.
The Frobenius inner product, sometimes denoted A : B, is the componentwise inner product of two matrices as though they are vectors. It is also the sum of the entries of the Hadamard product. Explicitly,
where "tr" denotes the trace of a matrix and vec denotes vectorization. This inner product induces the Frobenius norm.
For two matrices A and B of any different dimensions m×n and p×q respectively (no constraints on the dimensions of each matrix), the Kronecker product denoted A ⊗ B is a matrix with dimensions mp×nq, which has elements^{[citation needed]}
where represents the floor function.
Explicitly:
This is the application of the more general tensor product applied to matrices.
Wikimedia Commons has media related to matrix multiplication. 
The Wikibook The Book of Mathematical Proofs has a page on the topic of: Proofs of properties of matrices 
The Wikibook Linear Algebra has a page on the topic of: Matrix multiplication 
The Wikibook Applicable Mathematics has a page on the topic of: Multiplying Matrices 
