Positive-definite matrix

A positive-definite matrix is a square matrix whose associated quadratic expression is strictly positive for every nonzero vector. Under the standard convention in linear algebra, the matrix is symmetric when its entries are real and Hermitian when they are complex. Positive definiteness connects matrix algebra with geometry, optimization, and statistical modeling: it characterizes nondegenerate squared lengths and quadratic functions with a unique minimum. (mit.edu)

Definition and conventions

A real symmetric matrix $A\in\mathbb R^{n\times n}$ is positive definite if

Here $x^T$ denotes the transpose, and $q_A(x)=x^TAx$ is a quadratic form. The exclusion of the zero vector is essential, since $q_A(0)=0$. (mit.edu)

For a matrix with complex entries, the corresponding definition uses

where $A^*$ is the conjugate transpose. The condition $A=A^*$ defines a Hermitian matrix and ensures that $z^*Az$ is real. The notation $A\succ0$ commonly denotes positive definiteness. (mit.edu)

Some authors also apply the term to nonsymmetric real matrices satisfying $x^TAx>0$. This broader convention depends only on the symmetric part, because

The symmetric or Hermitian convention is used throughout the properties below. (web.mit.edu)

A positive-semidefinite matrix, written $A\succeq0$, permits equality for nonzero vectors. A positive-semidefinite matrix is positive definite exactly when it is invertible. Negative definiteness means that $-A$ is positive definite; an indefinite Hermitian matrix has quadratic-form values of both signs. (web.mit.edu)

Equivalent characterizations

For a symmetric or Hermitian matrix $A$, several conditions are equivalent:

• Positive eigenvalues: every eigenvalue of $A$ is strictly positive.
• Sylvester’s criterion: every leading principal minor is positive. These are the determinants of the upper-left $k\times k$ submatrices, for $k=1,\ldots,n$.
• Cholesky factorization: $A=LL^*$, where $L$ is lower triangular with strictly positive real diagonal entries.
• Gram representation: $A=B^*B$ for a matrix $B$ with linearly independent columns. (mit.edu)

The spectral theorem explains the first characterization. Writing $A=Q\Lambda Q^*$ gives

This is positive for every nonzero $z$ precisely when every $\lambda_i>0$. Likewise, $z^*B^*Bz=\|Bz\|_2^2$, which is positive precisely when $B$ has a trivial kernel. (web.mit.edu)

Sylvester’s criterion must not be weakened to checking only diagonal entries or the determinant. For example, $\operatorname{diag}(1,-1,-1)$ has positive determinant but is not positive definite. For positive semidefiniteness, nonnegative leading principal minors alone are insufficient; all principal minors must be nonnegative. (web.mit.edu)

Examples and geometric meaning

For

the leading principal minors are $2$ and $3$, and

for nonzero $x$. Thus $A$ is positive definite despite having negative off-diagonal entries. By contrast, the matrix with every entry equal to $1$ is positive semidefinite but singular: its quadratic form vanishes at $(1,-1)^T$. These examples illustrate that definiteness concerns the whole quadratic form, not entrywise positivity. (ocw.mit.edu)

A positive-definite matrix defines an inner product

and an associated norm, $\|x\|_A=\sqrt{x^*Ax}$. Conversely, the matrix representing an inner product in any finite-dimensional basis is positive definite. For real matrices, $x^TAx=1$ describes an ellipsoid whose principal directions are eigenvectors of $A$, with semiaxis lengths $1/\sqrt{\lambda_i}$. (cs.cornell.edu)

Algebraic structure and computation

The inverse of a positive-definite matrix is positive definite. Positive definiteness is also preserved by congruence transformations $A\mapsto S^*AS$ with invertible $S$, reflecting a change of coordinates. Every positive-definite matrix has a unique positive-definite square root, obtained by replacing its eigenvalues with their positive square roots. (web.mit.edu)

Positive-definite matrices form an open convex set within the real vector space of symmetric or Hermitian matrices. Their closure is the positive-semidefinite cone. In particular, positive weighted sums remain positive definite, and $A+\varepsilon I\succ0$ whenever $A\succeq0$ and $\varepsilon>0$, with $I$ the identity matrix. (web.stanford.edu)

In computation, Cholesky decomposition provides a factorization and a practical definiteness test. Once $A=LL^*$ is available, a linear system $Ax=b$ is solved through two triangular systems. Positive definiteness does not guarantee good conditioning: a very small minimum eigenvalue can make solutions sensitive to perturbations and rounding errors. (netlib.org)

Applications

In optimization, the quadratic function

has Hessian $A$. If $A\succ0$, it is strictly convex and has the unique minimizer $x=A^{-1}b$. More generally, a positive-definite Hessian at a stationary point of a twice continuously differentiable function guarantees a strict local minimum, though it is not necessary for one. (ocw.mit.edu)

In statistics, a covariance matrix $\Sigma$ is positive semidefinite because

It is positive definite exactly when every nonzero linear combination has positive variance. A nondegenerate multivariate normal distribution therefore has positive-definite covariance. Similarly, the Gram matrix $X^TX$ is positive definite exactly when $X$ has full column rank; adding $\lambda I$, with $\lambda>0$, makes it positive definite regardless of rank. (web.stanford.edu)