Positive definite matrices are the algebraic form of strictly positive curvature, squared length, covariance, and stable quadratic energy.
Positive definiteness
Definition. A Hermitian matrix \(A=A^*\) is positive semidefinite if \(x^*Ax\ge0\) for every \(x\), written \(A\succeq0\). It is positive definite if \(x^*Ax\gt0\) for every nonzero \(x\), written \(A\succ0\).
Example. For any matrix \(B\), the matrix \(B^*B\) is positive semidefinite because \(x^*B^*Bx=\|Bx\|^2\). It is positive definite exactly when \(B\) has trivial kernel, i.e. full column rank.
Equivalent criteria
Theorem. For a Hermitian matrix \(A\), the following are equivalent: \(A\succ0\); all eigenvalues of \(A\) are positive; there exists an invertible \(R\) with \(A=R^*R\); all leading principal minors are positive.
Proof. By the spectral theorem, \(x^*Ax\) is a weighted sum of eigenvalues in an orthonormal eigenbasis, so positivity of the quadratic form is equivalent to positivity of all eigenvalues. If eigenvalues are positive, let \(R=A^{1/2}\). Conversely \(R^*R\) is positive definite when \(R\) is invertible. The principal-minor criterion follows from Gaussian elimination without row exchanges, or equivalently from Cholesky factorization.
Cholesky factorization
Theorem. If \(A\succ0\), then there is a unique lower triangular matrix \(L\) with positive diagonal entries such that \[ A=LL^*. \] This is the Cholesky factorization.
Calculation. The first pivot is \(a_{11}\gt0\). Write \[ A=\begin{pmatrix}a&b^*\\b&C\end{pmatrix}. \] Eliminating the first column produces the Schur complement \(C-ba^{-1}b^*\), which remains positive definite. Repeating this step constructs \(L\).
Schur complements
Proposition. If \[ A=\begin{pmatrix}B&C\\C^*&D\end{pmatrix} \] with \(B\succ0\), then \(A\succ0\) if and only if \(D-C^*B^{-1}C\succ0\). The matrix \(D-C^*B^{-1}C\) is the Schur complement of \(B\).
Proof. Complete the square: \[ \begin{pmatrix}x\\y\end{pmatrix}^*A\begin{pmatrix}x\\y\end{pmatrix} =(x+B^{-1}Cy)^*B(x+B^{-1}Cy)+y^*(D-C^*B^{-1}C)y. \] Positivity of both terms is exactly the stated condition.
Covariance matrices and ellipsoids
Example. A covariance matrix \(\Sigma=\mathbb E[(X-\mu)(X-\mu)^*]\) is positive semidefinite because \(a^*\Sigma a=\operatorname{Var}(a^*X)\ge0\). If \(\Sigma\succ0\), the level sets \(x^*\Sigma^{-1}x=c\) are ellipsoids whose axes are eigenvectors of \(\Sigma\).
Counterexample. A matrix can have positive diagonal entries and still fail to be positive definite. For example, \[ \begin{pmatrix}1&2\\2&1\end{pmatrix} \] has eigenvalues \(3\) and \(-1\), so the quadratic form takes negative values.
Sylvester's criterion and Cholesky recursion
Theorem (Sylvester). A Hermitian \(n\times n\) matrix \(A\) is positive definite if and only if every leading principal minor \[ \Delta_k=\det A_{1:k,1:k} \] is positive.
Proof. If \(A\succ0\), every leading principal submatrix is positive definite and has positive eigenvalues, hence positive determinant. Conversely, \(\Delta_1>0\). Inductively write \[ A_k=\begin{pmatrix}A_{k-1}&b\\b^*&c\end{pmatrix}. \] By induction \(A_{k-1}\succ0\), and the determinant identity \[ \Delta_k=\Delta_{k-1}(c-b^*A_{k-1}^{-1}b) \] makes the Schur complement positive. Completing the square then proves \(A_k\succ0\).
The same induction computes Cholesky entries: \[ \ell_{jj} =\sqrt{a_{jj}-\sum_{k<j}|\ell_{jk}|^2},\qquad \ell_{ij} =\frac{a_{ij}-\sum_{k<j}\ell_{ik}\overline{\ell_{jk}}}{\ell_{jj}}. \] Positive definiteness guarantees that every square root is real and positive. This is both a proof of existence and a practical \(O(n^3)\) algorithm.