Definitiveness

A matrix is positive definite if $x^TAx > 0$, and positive semidefinite if $x^TAx \geq 0$.

PSD does not necessarily imply that it is symmetric, although we often restrict our discussions to symmetric matrices

These things are equivalent for PSD (and therefore are necessary & sufficient conditions)

• $x^TAx \geq 0$
  • element tool: use summation expansion
  • vector tool: look for norms

• Eigenvectors non-negative
  • vector tool: look for some norm contradiction
  • No guarantees that there is an eigenbasis though (although if it’s symmetric, it does have eigenbasis)

• $A = R^TR$ for some matrix $R$. Proof: take SVD, things cancel out, and you get a new SVD and each of the singular values are positive because they are squared). s

Other important properties for PSD (necessary, but not sufficient)

• invertible if positive definite (because there exists no vector that gets crushed to zero)

• If invertible (non-zero eigenvectors), the inverse of PSD is also PSD. Proof: assuming that it’s symmetric, you can invert it by taking the reciprocal of eigenvalues, which maintains positive

• determinant is always positive