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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.
In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs.
For example, the following 3×3 matrix is symmetric: [] Every square diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
Given a symmetric conference matrix, the matrix S can be viewed as the Seidel adjacency matrix of a graph. The graph has n − 1 vertices, corresponding to the rows and columns of S, and two vertices are adjacent if the corresponding entry in S is negative. This graph is strongly regular of the type called (after the matrix) a conference graph.
A hollow matrix may be a square matrix whose diagonal elements are all equal to zero. [3] That is, an n × n matrix A = (a ij) is hollow if a ij = 0 whenever i = j (i.e. a ii = 0 for all i). The most obvious example is the real skew-symmetric matrix. Other examples are the adjacency matrix of a finite simple graph, and a distance matrix or ...
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.
Null-symmetric matrix A square matrix whose null space (or kernel) is equal to its transpose, N(A) = N(A T) or ker(A) = ker(A T). Synonym for kernel-symmetric matrices. Examples include (but not limited to) symmetric, skew-symmetric, and normal matrices. Null-Hermitian matrix
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 ...