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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.
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]
The adjugate of a diagonal matrix is again diagonal. Where all matrices are square, 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.
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.
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.
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-diagonal matrix is also anti-diagonal, as can be seen from the paragraph above. The determinant of an anti-diagonal matrix has absolute value given by the product ...
the entries on the main diagonal are all zero (that is, the matrix is a hollow matrix), i.e. x ii = 0 for all 1 ≤ i ≤ N, all the off-diagonal entries are positive (x ij > 0 if i ≠ j), (that is, a non-negative matrix), the matrix is a symmetric matrix (x ij = x ji), and; for any i and j, x ij ≤ x ik + x kj for all k (the triangle ...