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A diagonal matrix where the diagonal elements are either +1 or −1. Single-entry matrix: A matrix where a single element is one and the rest of the elements are zero. Skew-Hermitian matrix: A square matrix which is equal to the negative of its conjugate transpose, A * = −A. Skew-symmetric matrix
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
Let A be an m × n matrix with real or complex entries. [a] If I is a subset of size r of {1, ..., m} and J is a subset of size s of {1, ..., n}, then the (I, J )-submatrix of A, written A I, J , is the submatrix formed from A by retaining only those rows indexed by I and those columns indexed by J.
A matrix is a rectangular array of numbers (or other mathematical objects), called the entries of the matrix. Matrices are subject to standard operations such as addition and multiplication. [2] Most commonly, a matrix over a field F is a rectangular array of elements of F.
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.
A band matrix with k 1 = k 2 = 0 is a diagonal matrix, with bandwidth 0. A band matrix with k 1 = k 2 = 1 is a tridiagonal matrix, with bandwidth 1. For k 1 = k 2 = 2 one has a pentadiagonal matrix and so on. Triangular matrices. For k 1 = 0, k 2 = n−1, one obtains the definition of an upper triangular matrix
The Hilbert matrix is also totally positive (meaning that the determinant of every submatrix is positive). The Hilbert matrix is an example of a Hankel matrix. It is also a specific example of a Cauchy matrix. The determinant can be expressed in closed form, as a special case of the Cauchy determinant. The determinant of the n × n Hilbert ...
An entity closely related to the covariance matrix is the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector , which can be written as = ( ()) ( ()), where is the matrix of the diagonal elements of (i.e., a diagonal matrix of the variances of for =, …,).