19
$\begingroup$
  1. How does one check whether symmetric $4\times4$ matrix is positive semi-definite?

  2. What if this matrix has also rank deficiency: is it rank 3?

  • 4
    You can use the determinant criterion: the upper-left $1\times 1$, $2\times 2$, $3\times 3$ and $4 \times 4$ squares should all have non-negative determinant.2011-05-23
  • 3
    But that determinant criterion isn't enough in general, as $\begin{bmatrix}0&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}$ shows.2011-05-23
  • 2
    @Yuval - I believe the determinant criterion holds for positive definite matrices but not necessarily for positive semidefinite ones. In other words, if some of the principal minors are zero, it does not necessarily imply the matrix is positive semidefinite.2011-05-23
  • 4
    In that case, you can add $\epsilon > 0$ (to the diagonal) and then rerun all your computations (just when the matrix doesn't have full rank). A suitable $\epsilon$ can be found by looking at the magnitude of the entries (we want to guarantee that we don't miss any small negative eigenvalue).2011-05-23
  • 0
    The matrix is positive semidefinite (and not strictly) iff the main determinant is zero, no?2016-02-03

8 Answers 8