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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 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.
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero.
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 (lower or upper) 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 .
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.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.