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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).
A numeric character reference refers to a character by its Universal Character Set/Unicode code point, and a character entity reference refers to a character by a predefined name. A numeric character reference uses the format &#nnnn; or &#xhhhh; where nnnn is the code point in decimal form, and hhhh is the code point in hexadecimal form.
In the case that A is identified with a Hermitian matrix, the matrix of A * is equal to its conjugate transpose. (If A is a real matrix, then this is equivalent to A T = A, that is, A is a symmetric matrix.) This condition implies that all eigenvalues of a Hermitian map are real: To see this, it is enough to apply it to the case when x = y is ...
Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the diagonal matrix (above), and therefore is uniquely determined by up to the order of its entries.) Essentially, the property of being symmetric for real matrices corresponds to the property of being Hermitian for ...
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 symmetric matrix can always be transformed in this way into a diagonal matrix which has only entries , + , along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of A {\displaystyle A} , i.e. it does not depend on the matrix S {\displaystyle S} used.