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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 ...
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.
Furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix. An anti-diagonal matrix is invertible if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero. The inverse of any invertible anti ...
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.
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.
By contrast, if most of the elements are non-zero, the matrix is considered dense. [1] The number of zero-valued elements divided by the total number of elements (e.g., m × n for an m × n matrix) is sometimes referred to as the sparsity of the matrix. Conceptually, sparsity corresponds to systems with few pairwise interactions.
Write the triangular matrix U as U = D + N, where D is diagonal and N is strictly upper triangular (and thus a nilpotent matrix). The diagonal matrix D contains the eigenvalues of A in arbitrary order (hence its Frobenius norm, squared, is the sum of the squared moduli of the eigenvalues of A, while the Frobenius norm of A, squared, is the sum ...
The trace of a matrix is the sum of the diagonal elements. The top-right to bottom-left diagonal is sometimes described as the minor diagonal or antidiagonal. The off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero. [4] [5]