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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:
Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, [2] to represent 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 ...
Note first that any 2 × 2 real matrix can be considered one of the three types of the complex number z = x + y ε, where ε 2 ∈ { −1, 0, +1 }. This z is a point on a complex subplane of the ring of matrices. [8] The case where the determinant is negative only arises in a plane with ε 2 =+1, that is a split-complex number plane. Only one ...
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.
The determinant of A, det(A), is equal to the triple product of x 0, x 1, and x 2 —the volume of the parallelepiped formed by the rows or columns: = (). The ...
In 1933, Raymond Paley discovered the Paley construction, which produces a Hadamard matrix of order q + 1 when q is any prime power that is congruent to 3 modulo 4 and that produces a Hadamard matrix of order 2(q + 1) when q is a prime power that is congruent to 1 modulo 4. [5]
In the mathematical field of linear algebra, an arrowhead matrix is a square matrix containing zeros in all entries except for the first row, first column, and main diagonal, these entries can be any number.