How does one check whether symmetric $4\times4$ matrix is positive semi-definite?
What if this matrix has also rank deficiency: is it rank 3?
How to check if a symmetric $4\times4$ matrix is positive semi-definite?
19
$\begingroup$
linear-algebra
matrices
-
4You 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
-
3But 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
-
4In 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
-
0The matrix is positive semidefinite (and not strictly) iff the main determinant is zero, no? – 2016-02-03