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In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by A T (among other notations). [1] The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. [2]
Vectorization (mathematics) In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a m × n matrix A, denoted vec ( A ), is the mn × 1 column vector obtained by stacking the columns of the matrix A on ...
Commutation matrix. In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec ( A) into vec ( AT ):
Definition. The adjugate of A is the transpose of the cofactor matrix C of A , In more detail, suppose R is a unital commutative ring and A is an n × n matrix with entries from R. The (i, j) - minor of A, denoted Mij, is the determinant of the (n − 1) × (n − 1) matrix that results from deleting row i and column j of A.
numpy .org. NumPy (pronounced / ˈnʌmpaɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [ 3] The predecessor of NumPy, Numeric, was originally created by Jim Hugunin with ...
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 ...
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.
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) .