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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 ...
If the matrix is symmetric indefinite, it may be still decomposed as = where is a permutation matrix (arising from the need to pivot), a lower unit triangular matrix, and is a direct sum of symmetric and blocks, which is called Bunch–Kaufman decomposition [6]
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.
Uniqueness: for positive definite matrices Cholesky decomposition is unique. However, it is not unique in the positive semi-definite case. Comment: if is real and symmetric, has all real elements; Comment: An alternative is the LDL decomposition, which can avoid extracting square roots.
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 table at the right shows two possibilities for 2-by-2 matrices.
The derivation of the result hinges on a few basic observations: The real matrix / /, with (), is well-defined and skew-symmetric.; Any skew-symmetric real matrix can be block-diagonalized via orthogonal real matrices, meaning there is () such that = with a real positive-definite diagonal matrix containing the singular values of .
In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form =, where is a unitary matrix and is a positive semi-definite Hermitian matrix (is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size.
The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite. When A is a real matrix (hence symmetric positive-definite), the factorization may be written =, where L is a real lower triangular matrix with positive diagonal entries.