Matrices With Explicitly Constrained Entries
The following lists matrices whose entries are subject to certain conditions. Many of them apply to square matrices only, that is matrices with the same number of columns and rows. The main diagonal of a square matrix is the diagonal joining the upper left corner and the lower right one or equivalently the entries ai,i. The other diagonal is called anti-diagonal (or counter-diagonal).
| Name | Explanation | Notes, References |
|---|---|---|
| (0,1)-matrix | A matrix with all elements either 0 or 1. | Synonym for binary matrix, Boolean matrix and logical matrix. |
| Alternant matrix | A matrix in which successive columns have a particular function applied to their entries. | |
| Anti-diagonal matrix | A square matrix with all entries off the anti-diagonal equal to zero. | |
| Anti-Hermitian matrix | Synonym for skew-Hermitian matrix. | |
| Anti-symmetric matrix | Synonym for skew-symmetric matrix. | |
| Arrowhead matrix | A square matrix containing zeros in all entries except for the first row, first column, and main diagonal. | |
| Band matrix | A square matrix whose non-zero entries are confined to a diagonal band. | |
| Bidiagonal matrix | A matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal. | Sometimes defined differently, see article. |
| Binary matrix | A matrix whose entries are all either 0 or 1. | Synonym for (0,1)-matrix, Boolean matrix or logical matrix. |
| Bisymmetric matrix | A square matrix that is symmetric with respect to its main diagonal and its main cross-diagonal. | |
| Block-diagonal matrix | A block matrix with entries only on the diagonal. | |
| Block matrix | A matrix partitioned in sub-matrices called blocks. | |
| Block tridiagonal matrix | A block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements | |
| Boolean matrix | A matrix whose entries are all either 0 or 1. | Synonym for (0,1)-matrix, binary matrix or logical matrix. |
| Cauchy matrix | A matrix whose elements are of the form 1/(xi + yj) for (xi), (yj) injective sequences (i.e., taking every value only once). | |
| Centrosymmetric matrix | A matrix symmetric about its center; i.e., aij = an−i+1,n−j+1 | |
| Conference matrix | A square matrix with zero diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix. | |
| Complex Hadamard matrix | A matrix with all rows and columns mutually orthogonal, whose entries are unimodular. | |
| Copositive matrix | A square matrix A with real coefficients, such that is nonnegative for every nonnegative vector x | |
| Diagonally dominant matrix | |aii| > Σj≠i |aij|. | |
| Diagonal matrix | A square matrix with all entries outside the main diagonal equal to zero. | |
| Elementary matrix | A square matrix derived by applying an elementary row operation to the identity matrix. | |
| Equivalent matrix | A matrix that can be derived from another matrix through a sequence of elementary row or column operations. | |
| Frobenius matrix | A square matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal. | |
| Generalized permutation matrix | A square matrix with precisely one nonzero element in each row and column. | |
| Hadamard matrix | A square matrix with entries +1, −1 whose rows are mutually orthogonal. | |
| Hankel matrix | A matrix with constant skew-diagonals; also an upside down Toeplitz matrix. | A square Hankel matrix is symmetric. |
| Hermitian matrix | A square matrix which is equal to its conjugate transpose, A = A*. | |
| Hessenberg matrix | An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal. | |
| Hollow matrix | A square matrix whose main diagonal comprises only zero elements. | |
| Integer matrix | A matrix whose entries are all integers. | |
| Logical matrix | A matrix with all entries either 0 or 1. | Synonym for (0,1)-matrix, binary matrix or Boolean matrix. Can be used to represent a k-adic relation. |
| Metzler matrix | A matrix whose off-diagonal entries are non-negative. | |
| Monomial matrix | A square matrix with exactly one non-zero entry in each row and column. | Synonym for generalized permutation matrix. |
| Moore matrix | A row consists of a, aq, aq², etc., and each row uses a different variable. | |
| Nonnegative matrix | A matrix with all nonnegative entries. | |
| Partitioned matrix | A matrix partitioned into sub-matrices, or equivalently, a matrix whose entries are themselves matrices rather than scalars | Synonym for block matrix |
| Parisi matrix | A block-hierarchical matrix. It consist of growing blocks placed along the diagonal, each block is itself a Parisi matrix of a smaller size. | In theory of spin-glasses is also known as a replica matrix. |
| Pentadiagonal matrix | A matrix with the only nonzero entries on the main diagonal and the two diagonals just above and below the main one. | |
| Permutation matrix | A matrix representation of a permutation, a square matrix with exactly one 1 in each row and column, and all other elements 0. | |
| Persymmetric matrix | A matrix that is symmetric about its northeast-southwest diagonal, i.e., aij = an−j+1,n−i+1 | |
| Polynomial matrix | A matrix whose entries are polynomials. | |
| Positive matrix | A matrix with all positive entries. | |
| Quaternionic matrix | A matrix whose entries are quaternions. | |
| Sign matrix | A matrix whose entries are either +1, 0, or −1. | |
| Signature matrix | A diagonal matrix where the diagonal elements are either +1 or −1. | |
| Skew-Hermitian matrix | A square matrix which is equal to the negative of its conjugate transpose, A* = −A. | |
| Skew-symmetric matrix | A matrix which is equal to the negative of its transpose, AT = −A. | |
| Skyline matrix | A rearrangement of the entries of a banded matrix which requires less space. | |
| Sparse matrix | A matrix with relatively few non-zero elements. | Sparse matrix algorithms can tackle huge sparse matrices that are utterly impractical for dense matrix algorithms. |
| Sylvester matrix | A square matrix whose entries come from coefficients of two polynomials. | The Sylvester matrix is nonsingular if and only if the two polynomials are coprime to each other. |
| Symmetric matrix | A square matrix which is equal to its transpose, A = AT (ai,j = aj,i). | |
| Toeplitz matrix | A matrix with constant diagonals. | |
| Triangular matrix | A matrix with all entries above the main diagonal equal to zero (lower triangular) or with all entries below the main diagonal equal to zero (upper triangular). | |
| Tridiagonal matrix | A matrix with the only nonzero entries on the main diagonal and the diagonals just above and below the main one. | |
| Unitary matrix | A square matrix whose inverse is equal to its conjugate transpose, A−1 = A*. | |
| Vandermonde matrix | A row consists of 1, a, a², a³, etc., and each row uses a different variable. | |
| Walsh matrix | A square matrix, with dimensions a power of 2, the entries of which are +1 or -1. | |
| Z-matrix | A matrix with all off-diagonal entries less than zero. |
Read more about this topic: List Of Matrices
Famous quotes containing the words explicitly and/or constrained:
“What happened at Hiroshima was not only that a scientific breakthrough ... had occurred and that a great part of the population of a city had been burned to death, but that the problem of the relation of the triumphs of modern science to the human purposes of man had been explicitly defined.”
—Archibald MacLeish (1892–1982)
“What a hell of an economic system! Some are replete with everything while others, whose stomachs are no less demanding, whose hunger is just as recurrent, have nothing to bite on. The worst of it is the constrained posture need puts you in. The needy man does not walk like the rest; he skips, slithers, twists, crawls.”
—Denis Diderot (1713–1784)