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Mathematical operation in linear algebra
For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The result matrix has the number of rows of the first and the number of columns of the second matrix.
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.
This article will use the following notational conventions: matrices are represented by capital letters in bold, e.g. A; vectors in lowercase bold, e.g. a; and entries of vectors and matrices are italic (since they are numbers from a field), e.g. A and a. Index notation is often the clearest way to express definitions, and is used as standard in the literature. The i, j entry of matrix A is indicated by (A)ij, Aij or aij, whereas a numerical label (not matrix entries) on a collection of matrices is subscripted only, e.g. A1, A2, etc.
If A is an m × n matrix and B is an n × p matrix,
the matrix productC = AB (denoted without multiplication signs or dots) is defined to be the m × p matrix
for i = 1, ..., m and j = 1, ..., p.
That is, the entry of the product is obtained by multiplying term-by-term the entries of the ith row of A and the jth column of B, and summing these n products. In other words, is the dot product of the ith row of A and the jth column of B.
Therefore, AB can also be written as
Thus the product AB is defined if and only if the number of columns in A equals the number of rows in B, in this case n.
In most scenarios, the entries are numbers, but they may be any kind of mathematical objects for which an addition and a multiplication are defined, that are associative, and such that the addition is commutative, and the multiplication is distributive with respect to the addition. In particular, the entries may be matrices themselves (see block matrix).
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:
Historically, matrix multiplication has been introduced for facilitating and clarifying computations in linear algebra. This strong relationship between matrix multiplication and linear algebra remains fundamental in all mathematics, as well as in physics, engineering and computer science.
If a vector space has a finite basis, its vectors are each uniquely represented by a finite sequence of scalars, called a coordinate vector, whose elements are the coordinates of the vector on the basis. These coordinate vectors form another vector space, which is isomorphic to the original vector space. A coordinate vector is commonly organized as a column matrix (also called column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space.
A linear mapA from a vector space of dimension n into a vector space of dimension m maps a column vector
onto the column vector
The linear map A is thus defined by the matrix
and maps the column vector to the matrix product
If B is another linear map from the preceding vector space of dimension m, into a vector space of dimension p, it is represented by a matrix A straightforward computation shows that the matrix of the composite map is the matrix product The general formula ) that defines the function composition is instanced here as a specific case of associativity of matrix product (see § Associativity below):
where denotes the conjugate transpose of (conjugate of the transpose, or equivalently transpose of the conjugate).
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, even when the product remains definite after changing the order of the factors.
An operation is commutative if, given two elements A and B such that the product is defined, then is also defined, and
If A and B are matrices of respective sizes and , then is defined if , and is defined if . Therefore, if one of the products is defined, the other is not defined in general. If , the two products are defined, but have different sizes; thus they cannot be equal. Only if , that is, if A and B are square matrices of the same size, are both products defined and of the same size. Even in this case, one has in general
This example may be expanded for showing that, if A is a matrix with entries in a fieldF, then for every matrix B with entries in F, if and only if where , and I is the identity matrix. If, instead of a field, the entries are supposed to belong to a ring, then one must add the condition that c belongs to the center of the ring.
One special case where commutativity does occur is when D and E are two (square) diagonal matrices (of the same size); then DE = ED. Again, if the matrices are over a general ring rather than a field, the corresponding entries in each must also commute with each other for this to hold.
The matrix product is distributive with respect to matrix addition. That is, if A, B, C, D are matrices of respective sizes m × n, n × p, n × p, and p × q, one has (left distributivity)
This results from the distributivity for coefficients by
Product with a scalar
If A is a matrix and c a scalar, then the matrices and are obtained by left or right multiplying all entries of A by c. If the scalars have the commutative property, then
If the product is defined (that is, the number of columns of A equals the number of rows of B), then
If the scalars have the commutative property, then all four matrices are equal. More generally, all four are equal if c belongs to the center of a ring containing the entries of the matrices, because in this case, cX = Xc for all matrices X.
These properties result from the bilinearity of the product of scalars:
If the scalars have the commutative property, the transpose of a product of matrices is the product, in the reverse order, of the transposes of the factors. That is
where T denotes the transpose, that is the interchange of rows and columns.
This identity does not hold for noncommutative entries, since the order between the entries of A and B is reversed, when one expands the definition of the matrix product.
This results from applying to the definition of matrix product the fact that the conjugate of a sum is the sum of the conjugates of the summands and the conjugate of a product is the product of the conjugates of the factors.
Transposition acts on the indices of the entries, while conjugation acts independently on the entries themselves. It results that, if A and B have complex entries, one has
where + denotes the conjugate transpose (conjugate of the transpose, or equivalently transpose of the conjugate).
Given three matrices A, B and C, the products (AB)C and A(BC) are defined if and only if the number of columns of A equals the number of rows of B, and the number of columns of B equals the number of rows of C (in particular, if one of the products is defined, then the other is also defined). In this case, one has the associative property
As for any associative operation, this allows omitting parentheses, and writing the above products as
This extends naturally to the product of any number of matrices provided that the dimensions match. That is, if A1, A2, ..., An are matrices such that the number of columns of Ai equals the number of rows of Ai + 1 for i = 1, ..., n - 1, then the product
These properties may be proved by straightforward but complicated summation manipulations. This result also follows from the fact that matrices represent linear maps. Therefore, the associative property of matrices is simply a specific case of the associative property of function composition.
Complexity is not associative
Although the result of a sequence of matrix products does not depend on the order of operation (provided that the order of the matrices is not changed), the computational complexity may depend dramatically on this order.
For example, if A, B and C are matrices of respective sizes 10×30, 30×5, 5×60, computing (AB)C needs 10×30×5 + 10×5×60 = 4,500 multiplications, while computing A(BC) needs 30×5×60 + 10×30×60 = 27,000 multiplications.
Algorithms have been designed for choosing the best order of products, see Matrix chain multiplication. When the number n of matrices increases, it has been shown that the choice of the best order has a complexity of
If n > 1, many matrices do not have a multiplicative inverse. For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix A is denoted A-1, and, thus verifies
A product of matrices is invertible if and only if each factor is invertible. In this case, one has
When R is commutative, and, in particular, when it is a field, the determinant of a product is the product of the determinants. As determinants are scalars, and scalars commute, one has thus
The other matrix invariants do not behave as well with products. Nevertheless, if R is commutative, and have the same trace, the same characteristic polynomial, and the same eigenvalues with the same multiplicities.
However, the eigenvectors are generally different if
Powers of a matrix
One may raise a square matrix to any nonnegative integer power multiplying it by itself repeatedly in the same way as for ordinary numbers. That is,
Computing the kth power of a matrix needs k - 1 times the time of a single matrix multiplication, if it is done with the trivial algorithm (repeated multiplication). As this may be very time consuming, one generally prefers using exponentiation by squaring, which requires less than 2 log2k matrix multiplications, and is therefore much more efficient.
An easy case for exponentiation is that of a diagonal matrix. Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the kth power of a diagonal matrix is obtained by raising the entries to the power k:
The definition of matrix product requires that the entries belong to a semiring, and does not require multiplication of elements of the semiring to be commutative. In many applications, the matrix elements belong to a field, although the tropical semiring is also a common choice for graph shortest path problems. Even in the case of matrices over fields, the product is not commutative in general, although it is associative and is distributive over matrix addition. The identity matrices (which are the square matrices whose entries are zero outside of the main diagonal and 1 on the main diagonal) are identity elements of the matrix product. It follows that the n × n matrices over a ring form a ring, which is noncommutative except if n = 1 and the ground ring is commutative.
Improvement of estimates of exponent ? over time for the computational complexity of matrix multiplication .
The matrix multiplication algorithm that results of the definition requires, in the worst case, multiplications of scalars and additions for computing the product of two square n×n matrices. Its computational complexity is therefore , in a model of computation for which the scalar operations require a constant time (in practice, this is the case for floating point numbers, but not for integers).
The greatest lower bound for the exponent of matrix multiplication algorithm is generally called . One has , because one has to read the elements of a matrix for multiplying it by another matrix. Thus . It is unknown whether . The largest known lower bound for matrix-multiplication complexity is ?(n2 log(n)), for a restricted kind of arithmetic circuits, and is due to Ran Raz.
The importance of the computational complexity of matrix multiplication relies on the facts that many algorithmic problems may be solved by means of matrix computation, and most problems on matrices have a complexity which is either the same as that of matrix multiplication (up to a multiplicative constant), or may be expressed in term of the complexity of matrix multiplication or its exponent
There are several advantages of expressing complexities in terms of the exponent of matrix multiplication. Firstly, if is improved, this will automatically improve the known upper bound of complexity of many algorithms. Secondly, in practical implementations, one never uses the matrix multiplication algorithm that has the best asymptotical complexity, because the constant hidden behind the big O notation is too large for making the algorithm competitive for sizes of matrices that can be manipulated in a computer. Thus expressing complexities in terms of provide a more realistic complexity, since it remains valid whichever algorithm is chosen for matrix computation.
Matrix inversion, determinant and Gaussian elimination
In his 1969 paper, where he proved the complexity for matrix computation, Strassen proved also that matrix inversion, determinant and Gaussian elimination have, up to a multiplicative constant, the same computational complexity as matrix multiplication. The proof does not make any assumptions on matrix multiplication that is used, except that its complexity is for some
The starting point of Strassen's proof is using block matrix multiplication. Specifically, a matrix of even dimension 2n×2n may be partitioned in four n×n blocks
Under this form, its inverse is
provided that A and are invertible.
Thus, the inverse of a 2n×2n matrix may be computed with two inversions, six multiplications and four additions or additive inverses of n×n matrices. It follows that, denoting respectively by I(n), M(n) and A(n) = n2 the number of operations needed for inverting, multiplying and adding n×n matrices, one has
If one may apply this formula recursively:
If and one gets eventually
for some constant d.
For matrices whose dimension is not a power of two, the same complexity is reached by increasing the dimension of the matrix to a power of two, by padding the matrix with rows and columns whose entries are 1 on the diagonal and 0 elsewhere.
This proves the asserted complexity for matrices such that all submatrices that have to be inverted are indeed invertible. This complexity is thus proved for almost all matrices, as a matrix with randomly chosen entries is invertible with probability one.
The same argument applies to LU decomposition, as, if the matrix A is invertible, the equality
defines a block LU decomposition that may be applied recursively to and for getting eventually a true LU decomposition of the original matrix.
The argument applies also for the determinant, since it results from the block LU decomposition that
Matrix calculus, for the interaction of matrix multiplication with operations from calculus
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