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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 ...
Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition. Singular matrices can also be factored, but not uniquely.
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.
If A and B are both symmetric or Hermitian, and B is also a positive-definite matrix, the eigenvalues λ i are real and eigenvectors v 1 and v 2 with distinct eigenvalues are B-orthogonal (v 1 * Bv 2 = 0). [15]
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,
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.
There exists a positive diagonal matrix D such that AD + DA T is positive definite. A is positive stable. That is, the real part of each eigenvalue of A is positive. There exists a symmetric positive definite matrix W such that AW + WA T is positive definite. A + I is non-singular, and G = (A + I) −1 (A − I) is convergent.