# List of matrices

Listed below are some important classes of matrices used in mathematics:

- (0,1)-matrix or
*binary matrix*- a matrix with all elements either 0 or 1. - Adjacency matrix - a symmetric (0,1)-matrix
- Anti-Hermitian matrix - another name for a
*skew-Hermitian matrix*. - Anti-symmetric matrix - another name for a
*skew-symmetric matrix*. - Block matrix
- Cartan matrix
- Cayley-Newbirth operation matrix
- Companion matrix - the companion matrix of a polynomial is a special form of matrix, whose eigenvalues are equal to the roots of the polynomial.
- Coxeter matrix
- Diagonal matrix - a square matrix with all entries off the main diagonal equal to zero.
- Diagonalizable matrix - a square matrix similar to a diagonal matrix. It has a complete set of linearly independent eigenvectors.
- Gell-Mann matrices
- Generalized permutation matrix - a square matrix with precisely one nonzero element in each row and column.
- Hadamard matrix
- Hankel matrix - a matrix with constant off 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.
- Hessian matrix
- Hilbert matrix - a Hankel matrix with elements
*H*_{ij}= (i + j - 1)^{-1}. - Identity matrix - a square diagonal matrix, with all entries on the main diagonal equal to 1.
- Invertible matrix - a square matrix with a multiplicative inverse.
- Matrix code
- Matrix exponential - defined by the exponential series
- Matrix representation of conic sections
- Nonnegative matrix - a matrix with all nonnegative entries.
- Normal matrix - a square matrix that commutes with its conjugate transpose. Normal matrices are precisely the matrices to which the spectral theorem applies.
- Orthogonal matrix - a matrix whose inverse is equal to its transpose,
*A*^{−1}=*A*^{T}. - Orthonormal matrix
- Payoff matrix
- Permutation matrix - matrix representation of a permutation.
- Persymmetric matrix - a matrix that is symmetric about its northeast-southwest diagonal, i.e.,
*a*_{ij}=a_{n-j+1,n-i+1} - Pick matrix - occurs in the study of analytical interpolation problems
- Positive-definite matrix - a Hermitian matrix with every eigenvalue positive.
- Positive matrix - a matrix with all positive entries.
- S matrix - in physics
- Singular matrix - a noninvertible square matrix.
- Square matrix - an
*n*by*n*matrix. The set of all square matricies form an associative algebra with identity. - 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,
*A*^{T}= −*A*. - Stochastic matrix - a positive matrix describing a stochastic process. The sum of entries of any row is one.
- Substitution matrix
- Symmetric matrix - a square matrix which is equal to its transpose,
*A*=*A*^{T}. - Symplectic matrix - a square matrix preserving a standard skew-symmetric form.
- Toeplitz matrix - a matrix with constant diagonals.
- Totally positive matrix - a matrix with determinants of all its square submatrices positive. It is used in generating the reference points of Bézier curve in computer graphics.
- Totally unimodular matrix - a matrix for which every non-singular square submatrix is unimodular. This has some implications in the linear programming relaxation of an integer program.
- Transformation matrix
- Transition matrix - a matrix representing the probabilities of changing from one state to another
- 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.
- Unimodular matrix - a square matrix with determinant +1 or -1.
- Unitary matrix - a square matrix whose inverse is equal to its conjugate transpose,
*A*^{−1}=*A*^{*}. - Vandermonde matrix - a row consists of 1,
*a*,*a*^{2},*a*^{3}, etc., and each row uses a different variable - Walsh matrix
- Wronskian
- Zero matrix - a matrix with all entries equal to zero.