In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices...

linear algebra, a square matrix A {\displaystyle A}  is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists...

A block diagonal matrix is a block matrix that is a square matrix such that the main-diagonal blocks are square matrices and all off-diagonal blocks are...

tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this)...

anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal). An n-by-n matrix A is...

following equivalent conditions. M {\displaystyle M} is congruent with a diagonal matrix with positive real entries. M {\displaystyle M} is symmetric or Hermitian...

Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of...

spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation...

triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly...

algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A {\displaystyle...

the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance...

triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix. The identity matrix In of size n is the n-by-n matrix in...

matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal...

semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). The covariance matrix of a random vector X {\displaystyle...

of j-th column of the matrix A. Yet, there is a special basis for an operator in which the components form a diagonal matrix and, thus, multiplication...

matrix that describes adjacency in bipartite graphs. Degree matrix — a diagonal matrix defining the degree of each vertex in a graph. Edmonds matrix —...

where Q is the square n × n matrix whose ith column is the eigenvector qi of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding...

matrices, the diagonal matrix, is to store just the entries in the main diagonal as a one-dimensional array, so a diagonal n × n matrix requires only...

m} ⁠ complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is an m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real...

the main diagonal are called diagonal matrices. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix A = [ 1...

long as diagonally-opposite entries are complex conjugates. A matrix that has only real entries is symmetric if and only if it is a Hermitian matrix. A real...

lower triangular with ones on the diagonal, U is upper triangular with ones on the diagonal, and D is a diagonal matrix. Related: the LUP decomposition...

transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing...

scalar multiple of a skew-symmetric matrix is skew-symmetric. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace...

also called a heavy node, results in a large diagonal entry in the Laplacian matrix dominating the matrix properties. Normalization is aimed to make the...

the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the kth power of a diagonal matrix is obtained by...

adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is...

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;...

mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is greater...

U^{*}U=UU^{*}} ). U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, U has a decomposition...