Symmetric positive-definite matrix

Definition

A real $n\times n$ matrix $A$ is symmetric positive definite if $A^T=A$ and

v^TAv>0

for every nonzero $v\in\mathbb R^n$ . The set of such matrices is often denoted $\operatorname{Sym}_{++}(n,\mathbb R)$ .

Equivalent characterizations

For a real symmetric matrix, positive definiteness is equivalent to all eigenvalues being positive. It is also equivalent to the existence of an invertible matrix $B$ with $A=B^TB$ , and to positivity of all leading principal minors.

Geometry of the cone

The symmetric positive-definite matrices form an open convex cone in the vector space of real symmetric matrices. A Riemannian metric in local coordinates is a smoothly varying matrix with values in this cone.

References

Roger A. Horn and Charles R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2012. Publisher record. Relevant: positive-definite matrices.