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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
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let Mat n {\displaystyle {\mbox{Mat}}_{n}} denote the space of n × n {\displaystyle n\times n} matrices.
Matrix types (special types like bidiagonal/tridiagonal are not listed): Real – general (nonsymmetric) real; Complex – general (nonsymmetric) complex; SPD – symmetric positive definite (real) HPD – Hermitian positive definite (complex) SY – symmetric (real) HE – Hermitian (complex) BND – band; Operations: TF – triangular ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. The sum of two skew-symmetric matrices is skew-symmetric. A scalar multiple of a skew-symmetric matrix is skew-symmetric. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero.
Symmetry pattern of a bisymmetric 5 × 5 matrix. In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = A T (it is its own transpose), and AJ = JA, where J is the n × n exchange matrix. For example, any matrix of the form
The inverse of a Lehmer matrix is a tridiagonal matrix, where the superdiagonal and subdiagonal have strictly negative entries. Consider again the n×n A and m×m B Lehmer matrices, where m > n . A rather peculiar property of their inverses is that A −1 is nearly a submatrix of B −1 , except for the A −1 n,n element, which is not equal to ...
Any symmetric permutation of the arrowhead matrix, , where P is a permutation matrix, is a (permuted) arrowhead matrix. Real symmetric arrowhead matrices are used in some algorithms for finding of eigenvalues and eigenvectors .