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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, chemistry, engineering and computer science.
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 ...
The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, vec ( A B C ) = ( C T ⊗ A ) vec ( B ) {\displaystyle \operatorname {vec} (ABC)=(C^{\mathrm {T} }\otimes A)\operatorname {vec} (B)} for matrices A , B , and C of dimensions k ...
It is called an identity matrix because multiplication with it leaves a matrix unchanged: = = for any m-by-n matrix A. A nonzero scalar multiple of an identity matrix is called a scalar matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group ...
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.
Using the definition of trace as the sum of diagonal elements, the matrix formula tr(AB) = tr(BA) is straightforward to prove, and was given above. In the present perspective, one is considering linear maps S and T , and viewing them as sums of rank-one maps, so that there are linear functionals φ i and ψ j and nonzero vectors v i and w j ...
This formula simplifies significantly when the upper right block matrix B is the zero matrix. This formulation is useful when the matrices A and D have relatively simple inverse formulas (or pseudo inverses in the case where the blocks are not all square. In this special case, the block matrix inversion formula stated in full generality above ...