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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 ...
The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in multivariate analysis , where the sample covariance matrices are PSD.
Equivalently, the eigenvalues of are positive, and this implies that > since the determinant is the product of the eigenvalues. To prove the reverse implication, we use induction . The general form of an ( n + 1 ) × ( n + 1 ) {\displaystyle (n+1)\times (n+1)} Hermitian matrix is
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.
A positive-semidefinite matrix P can be written as N * N, ... To prove (1), consider P =M * M where M * is the conjugate transpose of M, and let the eigenvalues of P ...
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 is not necessarily positive semidefinite. For example, consider
The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can be seen from the following simple derivation:
Uniqueness: is always unique and equal to (which is always hermitian and positive semidefinite). If A {\displaystyle A} is invertible, then U {\displaystyle U} is unique. Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary matrix, P {\displaystyle P} can be written as P = V D V ∗ {\displaystyle P=VDV^{*}} .