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In mathematics , matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field , or, more generally, in a ring. The matrix product is designed for representing the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra , and as such has numerous applications in many areas of mathematics, as well as in applied mathematics , physics , and engineering.
When two linear maps are represented by matrices, then the matrix product represents the composition of the two maps. The definition of matrix product requires that the entries belong to a ring, which may be noncommutative , but is a field in most applications. Even in this latter case, matrix 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 all entries are zero, except those of the main diagonal that are all equal to 1 are identity elements of the matrix product.
A square matrix may have a multiplicative inverse , called an inverse matrix. In the common case where the entries belong to a commutative ring r , a matrix has an inverse if and only if its determinant has a multiplicative inverse in r. The determinant of a product of square matrices is the product of the determinants of the factors. Many classical groups including all finite groups are isomorphic to matrix groups; this is the starting point of the theory of group representations.
Computing matrix products is a central operation in all computational applications of linear algebra. This nonlinear complexity means that matrix product is often the critical part of many algorithms.
This is enforced by the fact that many operations on matrices, such as matrix inversion, determinant, solving systems of linear equations , have the same complexity. Therefore various algorithms have been devised for computing products of large matrices, taking into account the architecture of computers see BLAS , for example. This article will use the following notational conventions: A , vectors in lowercase bold, e. 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 or A ij , whereas a numerical label not matrix entries on a collection of matrices is subscripted only, e. 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 m. Usually the entries are numbers, but they may be any kind 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. Historically, matrix multiplication has been introduced for making easier 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 elements vectors are uniquely represented by a finite sequence , called coordinate vector , or scalars, which are the coordinates of the vector on the basis.
These coordinates are commonly organized as a column matrix also called column vector , that is a matrix with only one column. A linear map A from a vector space of dimension n into a vector space of dimension m maps a column vector. The linear map A is thus defined by the matrix. The general form of a system of linear equations is.
Using same notation as above, such a system is equivalent with the single matrix equation. The dot product of two column vectors is the matrix product. More generally, any bilinear form over a vector space of finite dimension may be expressed as a matrix product. 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.
Even in this case, one has in general. 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. The matrix product is distributive with respect of matrix addition. If the scalars have the commutative property, then all four matrices are equal.
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. 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. If A and B have complex entries, then. This results of 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. Given three matrices A , B and C , the products AB C and A BC are defined if and only 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 product is defined, 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 A B C.
This extends naturally to the product of any number of matrix provided that the dimension match. These properties may be proved by straightforward but complicate summation manipulations.
This result also 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. Although the result of a sequence of matrix product 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.
Algorithms have been designed for choosing the best order of products, see Matrix chain multiplication. This ring is also an associative R -algebra.
For example, a matrix such that all entries of a row or a column are 0 does not have an inverse. A matrix that has an inverse is an invertible matrix. Otherwise, it is a singular matrix. 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.
As for any ring, one may raise a square matrix to any nonnegative integer power multiplying it by itself repeatedly in the same way as for ordinary numbers. Computing the k th 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 log 2 k 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 k th power of a diagonal matrix is obtained by raising the entries to the power k:. 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 thant can be manipulated in a computer.
Problems that have the same asymptotic complexity as matrix multiplication include determinant , matrix inversion , Gaussian elimination see next section. In his paper, where he proved the complexity O n 2. The starting point of Strassen's proof is using block matrix multiplication. 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 all entries are zero but those on the main diagonal that are equal one.
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. The argument applies also for the determinant, since it results from the block LU decomposition that.
The term "matrix multiplication" is most commonly reserved for the definition given in this article. It could be more loosely applied to other operations on matrices. From Wikipedia, the free encyclopedia. For differentials and derivatives of products of matrices, see matrix calculus. For implementation technics in particular parallel and distributed algorithms , see Matrix multiplication algorithm.
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