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Block matrix. 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 ...
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: Input: matrices A and B.
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the ...
Kronecker product. In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a ...
Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix : − − where is given by the companion matrix. The set of × circulant matrices forms an - dimensional vector space with respect to addition and scalar multiplication. This space can be interpreted as the space of functions on the cyclic ...
By the formulas above, those n × n permutation matrices form a group of order n! under matrix multiplication, with the identity matrix as its identity element, a group that we denote . The group P n {\displaystyle {\mathcal {P}}_{n}} is a subgroup of the general linear group G L n ( R ) {\displaystyle GL_{n}(\mathbb {R} )} of invertible n × n ...
Calculus. In vector calculus, the Jacobian matrix ( / dʒəˈkoʊbiən /, [ 1][ 2][ 3] / dʒɪ -, jɪ -/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components ...
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃəˈlɛski / 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.