Definite matrix

A Hermitian matrix $A$ is definite when either $x^{†}Ax>0$ for every nonzero vector $x$, or $x^{†}Ax<0$ for every nonzero vector $x$. These are the positive-definite matrix and negative-definite matrix cases. By the spectral theorem, all eigenvalues must have the same strict sign, so every definite matrix is invertible.

A positive semidefinite matrix permits zero eigenvalues and need not be definite. For example, $\mathrm{diag}(1,0)$ is positive semidefinite but not definite, while $\mathrm{diag}(1,-1)$ is an indefinite matrix. Definiteness concerns the sign on every nonzero vector, not merely one diagonal entry or the determinant.

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.