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The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero. If is a real skew-symmetric matrix and is a real eigenvalue, then =, i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real. If is a real skew-symmetric matrix, then + is invertible, where is the identity matrix.
The matrix is obtained by taking for S the q × q matrix that has a +1 in position (i, j ) and −1 in position (j, i) if there is an arc of the digraph from i to j, and zero diagonal. Then C constructed as above from S, but with the first row all negative, is a skew-symmetric conference 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.
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.
The Pfaffian of a n × n skew-symmetric matrix for n odd is defined to be zero, as the determinant of an odd skew-symmetric matrix is zero, since for a skew-symmetric matrix, = = = (), and for n odd, this implies =.
If instead, A is equal to the negative of its transpose, that is, A = −A T, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfies A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex ...
Sometimes the notation is used instead of for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a complex structure , which often has the same coordinate expression as Ω {\displaystyle \Omega } but represents a very different structure.
In linear algebra, a skew-Hamiltonian matrix is a specific type of matrix that corresponds to a skew-symmetric bilinear form on a symplectic vector space. Let be a vector space equipped with a symplectic form, denoted by Ω. A symplectic vector space must necessarily be of even dimension.