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Conjugate transpose. In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , [1] , [2] or (often in physics) .
If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.
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.
Block matrix. hide. 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 ...
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m, then their outer product is an n × m matrix. More generally, given two tensors (multidimensional arrays of ...
Some authors use the name square root or the notation A1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = BTB = A (for real-valued matrices, where BT is the transpose of B ). Less frequently, the name square root may be used for any factorization of a ...
Adjugate matrix. In linear algebra, the adjugate of a square matrix A is the transpose of its cofactor matrix and is denoted by adj (A). [1] [2] It is also occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the conjugate ...
In-place matrix transposition. In-place matrix transposition, also called in-situ matrix transposition, is the problem of transposing an N × M matrix in-place in computer memory, ideally with O (1) (bounded) additional storage, or at most with additional storage much less than NM. Typically, the matrix is assumed to be stored in row-major or ...