A definite matrix is a Hermitian matrix whose associated quadratic form has a fixed sign. Positive and negative semidefinite matrices allow zero as well.
A real symmetric or complex Hermitian matrix is positive semidefinite when for every vector , equivalently when all its eigenvalues are nonnegative.
If is real symmetric and positive semidefinite, its principal square root isIt is symmetric and positive semidefinite, and is positive definite exactly when is.
Every invertible real matrix has the polar decompositionwhere is symmetric positive definite and is orthogonal.
Every invertible complex matrix has the unique polar decompositionwhere is a positive-definite Hermitian matrix and is a unitary matrix. Writing gives , which is Hermitian.
A real symmetric matrix is positive definite whenfor every nonzero real vector . It therefore defines the inner product and is invertible.
Every real symmetric positive-definite matrix has a unique factorization in which is a lower triangular matrix with positive diagonal entries.
Given a Cholesky decomposition , a rank-one Cholesky update computes a triangular factor of in operations without refactorizing the matrix from scratch.
A Hermitian matrix is positive definite exactly when all its leading principal minors are positive.
The condition number of a problem measures how strongly relative perturbations of its input can amplify relative changes in its output. For an invertible matrix and a chosen operator norm, .
For a real symmetric positive-definite matrix,It measures the sensitivity of the linear system and controls convergence bounds for iterative solvers.
An symmetric tridiagonal Toeplitz matrix has one constant diagonal and equal constant subdiagonal and superdiagonal . Its eigenvectors are the discrete sine transform vectors , with eigenvalues .
An Toeplitz antisymmetric tridiagonal matrix with diagonal , superdiagonal , and subdiagonal has eigenvaluesAll such matrices share an orthonormal eigenbasis after the same diagonal unitary change from the discrete sine basis.
A real symmetric matrix is negative definite when for every nonzero real vector . Equivalently, is a positive-definite matrix and every eigenvalue of is negative.
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In linear algebra, a definite matrix refers to a square matrix that has specific properties related to the positivity of its quadratic forms. The terminology typically includes several definitions: 1. **Positive Definite Matrix**: A symmetric matrix \( A \) is called positive definite if for all non-zero vectors \( x \), the following holds: \[ x^T A x > 0. \] This implies that all eigenvalues of the matrix are positive.
