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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.
For a square matrix, the diagonal (or main diagonal or principal diagonal) is the diagonal line of entries running from the top-left corner to the bottom-right corner. [ 1 ] [ 2 ] [ 3 ] For a matrix A {\displaystyle A} with row index specified by i {\displaystyle i} and column index specified by j {\displaystyle j} , these would be entries A i ...
The binary matrix with ones on the anti-diagonal, and zeroes everywhere else. a ij = δ n+1−i,j: A permutation matrix. Hilbert matrix: a ij = (i + j − 1) −1. A Hankel matrix. Identity matrix: A square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0. a ij = δ ij: Lehmer matrix: a ij = min(i, j) ÷ max(i, j).
In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edges attached to each vertex. [1]
A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix [2]) is non-singular. A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semidefinite. This follows from the eigenvalues being real, and Gershgorin's circle theorem. If the symmetry requirement is eliminated, such a matrix ...
Diagonal system of control, the method referees and assistant referees use to position themselves in one of the four quadrants of an association football pitch; Main diagonal, the entries of a matrix whose row and column indices are the same; Diagonal matrix, a matrix whose entries off the main diagonal are all zero
The spectral decomposition is a special case of the singular value decomposition, which states that any matrix can be expressed as = , where and are unitary matrices and is a diagonal matrix. The diagonal entries of Σ {\displaystyle \ \Sigma \ } are uniquely determined by A {\displaystyle \ A\ } and are known as the singular values of A ...
The remaining task has been reduced to finding the eigenvalues of a diagonal matrix plus a rank-one correction. Before showing how to do this, let us simplify the notation. We are looking for the eigenvalues of the matrix D + w w T {\displaystyle D+ww^{T}} , where D {\displaystyle D} is diagonal with distinct entries and w {\displaystyle w} is ...