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The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
The left column visualizes the calculations necessary to determine the result of a 2x2 matrix multiplication. Naïve matrix multiplication requires one multiplication for each "1" of the left column. Each of the other columns (M1-M7) represents a single one of the 7 multiplications in the Strassen algorithm. The sum of the columns M1-M7 gives ...
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, [10] even when the product remains defined after changing the order of the factors. [11] [12]
One can keep track of this fact by declaring an matrix to be of type , and similarly a matrix to be of type . This way, when q = n {\displaystyle q=n} the two arrows have matching source and target, m → n → p {\displaystyle m\to n\to p} , and can hence be composed to an arrow of type m → p {\displaystyle m\to p} .
The set M(n, R) (also denoted M n (R) [7]) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module R n. [58] If the ring R is commutative , that is, its multiplication is commutative, then the ring M( n , R ) is also an associative algebra over R .
When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using general inversion algorithms or by performing inverse operations (that have obvious geometric interpretation, like rotating ...
If A is a real m×n matrix, then det(A A T) is equal to the square of the m-dimensional volume of the parallelotope spanned in R n by the m rows of A. Binet's formula states that this is equal to the sum of the squares of the volumes that arise if the parallelepiped is orthogonally projected onto the m-dimensional coordinate planes (of which ...
In this representation the matrix () squares to the zero matrix, corresponding to the dual number . There are other ways to represent dual numbers as square matrices. They consist of representing the dual number 1 {\displaystyle 1} by the identity matrix , and ε {\displaystyle \varepsilon } by any matrix whose square is the zero matrix; that ...