Determine the conditions on $a$, $b$, and $c$ such that $\mathsf{M}$ is positive semidefinite

Question

Problem statement:

Given the matrix

$\mathsf{M} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ a & b & c & b & a \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$,

what are the conditions on $a$, $b$, and $c$ such that $\mathsf{M}$ is positive semidefinite?

Solution attempt:

The eigenvalues of $\mathsf{M}$ are given by

$\begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ c \end{bmatrix}$.

Does this mean that $c \geq 0$ for $\mathsf{M}$ to be positive semidefinite, independent of the values of $a$ and $b$?

Answer 1

$M$ is PSD if and only if the symmetric matrix $M + M^T$ has non-negative eigenvalues. In this case, we find that $M$ is PSD if and only if $c \geq 0$ and $a = b = 0$.

Note in particular that $M + M^T$ has the principal submatrices $$ \pmatrix{0&a\\a&2c}, \quad \pmatrix{0&b\\b&2c} $$ The first matrix is indefinite whenever $a \neq 0$, and the second is indefinite whenever $b \neq 0$.