# Why is it important to know whether matrix is positive definite?

There is a lot of info on them; also several ways to check for the property.

What I am missing is the application. Why is it important to know whether a matrix (a linear map) is positive definite?

• – Michael Hoppe Aug 29 '17 at 10:50
• @Michael Hoppe Thanks -- so it seems quite useful in gradient-based optimisation, to check if a point is a minimum (sounds like a stopping criterion). – A.L. Verminburger Aug 29 '17 at 10:57
• If $A$ is any matrix with real entries, then $A^T A$ is positive semidefinite. Furthermore, if $C$ is a positive semidefinite matrix (for example, a diagonal matrix with positive entries), then $A^T C A$ is positive semidefinite. This pattern $A^T C A$ tends to recur throughout math. Even the Laplace operator has this form. The ubiquity of $A^T C A$ in applied math is emphasized by Gilbert Strang in his textbooks. He gives many applications and examples where $A^T C A$ appears. Pos. definite matrices have nice properties, e.g., they are orthogonally diagonalizable with positive eigenvalues. – littleO Aug 29 '17 at 11:01

## 2 Answers

Let $B$ be a symmetric bilinear form defined in $\mathbb{R}^n\times\mathbb{R}^n$ and let $M_B$ be the matrix of $B$ with respect to the standard basis (that is, the entries of $M_B$ are the $B(e_i,e_j)$'s). Then $B$ is an inner product if and only if $M_B$ is positive definite.

If one has function $f$ on $\mathbb R$ and wants to know if it is the Fourier transform of a finite measure, a very convenient test is that all the matrices $M=(M_{i,j})$ with entries of form $M_{i,j} = f(x_i-x_j)$ must be positive definite.