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A matrix whose entries are either +1, 0, or −1. Signature matrix: A diagonal matrix where the diagonal elements are either +1 or −1. Single-entry matrix: A matrix where a single element is one and the rest of the elements are zero. Skew-Hermitian matrix: A square matrix which is equal to the negative of its conjugate transpose, A * = −A.
A matrix is diagonal if and only if it is triangular and normal. A matrix is diagonal if and only if it is both upper-and lower-triangular. A diagonal matrix is symmetric. The identity matrix I n and zero matrix are diagonal. A 1×1 matrix is always diagonal. The square of a 2×2 matrix with zero trace is always diagonal.
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix is the list of entries , where =. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones:
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 ...
An atomic (upper or lower) triangular matrix is a special form of unitriangular matrix, where all of the off-diagonal elements are zero, except for the entries in a single column. Such a matrix is also called a Frobenius matrix , a Gauss matrix , or a Gauss transformation matrix .
The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero. If A {\textstyle A} is a real skew-symmetric matrix and λ {\textstyle \lambda } is a real eigenvalue , then λ = 0 {\textstyle \lambda =0} , i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator [1] represented in an orthonormal basis over a real inner product space.
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal.