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The matrix multiplication algorithm that results from the definition requires, in the worst case, multiplications and additions of scalars to compute the product of two square n×n matrices.
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:
A square matrix of order 4. The entries form the main diagonal of a square matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a 11 = 9, a 22 = 11, a 33 = 4, a 44 = 10. In mathematics, a square matrix is a matrix with the same number of rows and columns.
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical ...
The sum of the columns M1-M7 gives the same result as the full matrix multiplication on the left. Let , be two square matrices over a ring, for example matrices whose entries are integers or the real numbers. The goal of matrix multiplication is to calculate the matrix product =.
Square root of 2, Newton's method (()) Euler's number, Binary splitting of the Taylor series for the exponential function (() ... Matrix multiplication One ...
The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above, the trace of the product of two matrices is independent of the order of the factors: = ().
In this case, we say that A and B are conformable for multiplication (in that sequence). Since squaring a matrix involves multiplying it by itself (A 2 = AA) a matrix must be m × m (that is, it must be a square matrix) to be conformable for squaring. Thus for example only a square matrix can be idempotent.