Positive semidefinite matrix - Imported Knowlpedia Content

Definition

Let $A\in M_n(\mathbb F)$ , where $\mathbb F=\mathbb R$ or $\mathbb C$ . The matrix $A$ is positive semidefinite, written $A\succeq0$ or $A\geq0$ , if $A=A^*$ and

x^*Ax\geq0\qquad\text{for every }x\in\mathbb F^n.

Here $A^*$ is the transpose in the real case and the conjugate transpose in the complex case. It is positive definite if $x^*Ax>0$ for every nonzero $x$ . Semidefiniteness permits a nontrivial kernel; definiteness does not. The condition is basis-independent when $A$ is regarded as the matrix of a self-adjoint operator on a finite-dimensional inner-product space.

Equivalent characterizations

For a Hermitian matrix $A$ , the following are equivalent:

$A$ is positive semidefinite;
every eigenvalue of $A$ is nonnegative;
there is a matrix $B$ such that $A=B^*B$ ;
$A$ has a unique positive semidefinite square root $A^{1/2}$ .

The factorization can be obtained from the spectral theorem. It also identifies positive semidefinite matrices with Gram matrices: $A_{ij}=\langle v_j,v_i\rangle$ for some finite family of vectors, subject to the convention chosen for which inner-product variable is linear.

Cone and order

Positive semidefinite matrices form a closed convex cone: if $A,B\succeq0$ and $s,t\geq0$ , then $sA+tB\succeq0$ . The Loewner order on Hermitian matrices is

A\preceq B\quad\Longleftrightarrow\quad B-A\succeq0.

This is a partial order, but it is not a total order when $n>1$ . Congruence preserves positivity: $A\succeq0$ implies $C^*AC\succeq0$ for every compatible matrix $C$ .

Tests and examples

Every matrix $B^*B$ is positive semidefinite. Covariance and Gram matrices are standard examples. A diagonal Hermitian matrix is positive semidefinite exactly when all diagonal entries are nonnegative.

Nonnegative entries do not imply positive semidefiniteness, and positive semidefinite matrices may have negative off-diagonal entries. Sylvester's criterion using strictly positive leading principal minors characterizes positive definiteness; for semidefiniteness one instead requires all principal minors to be nonnegative.

References

Roger A. Horn and Charles R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013. DOI record. Relevant: Chapters 4 and 7 on Hermitian matrices, quadratic forms, and positive semidefinite matrices.
Rajendra Bhatia, Positive Definite Matrices, Princeton University Press, 2007. DOI record. Relevant: Chapter 1 on positivity, factorization, and the Loewner order.