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A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if is a complex symmetric matrix, there is a unitary matrix such that is a real diagonal matrix with non-negative entries.
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
If a matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is invertible and its inverse is given by = If is a symmetric matrix, since is formed from the eigenvectors of , is guaranteed to be an orthogonal matrix, therefore =.
Orthostochastic matrix — doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix; Precision matrix — a symmetric n×n matrix, formed by inverting the covariance matrix. Also called the information matrix. Stochastic matrix — a non-negative matrix describing a stochastic ...
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. [1] [2] ... is symmetric and idempotent, ...
If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.
A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible. [31]
The matrix exponential of a real symmetric matrix is positive definite. Let be an n×n real symmetric matrix and a column vector. Using the elementary properties of the matrix exponential and of symmetric matrices, we have: