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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.
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 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 ...
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.
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]
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.
Thus the square roots of A are given by VD 1/2 V −1, where D 1/2 is any square root matrix of D, which, for distinct eigenvalues, must be diagonal with diagonal elements equal to square roots of the diagonal elements of D; since there are two possible choices for a square root of each diagonal element of D, there are 2 n choices for the ...