Parameters that will make this matrix positive semidefinite

Question

I have a matrix $$M=\begin{bmatrix} 1+t+m &n&t+n&m+c \\ n &1+t-m&m-c & t-n \\ t+n & m-c&1-t-m & -n \\ m+c & t-n & -n & 1-t+m \end{bmatrix}$$ where I know that $0 \leq c \leq 1$ and $ t=a+(m+n)b$ for some fixed $0 \leq a,b\leq 1$. Here $m$ and $n$ are free parameters with $t$ depending on $m,n$. I'm trying to find a pair of real numbers $(m,n)$ which ensure that $M$ is positive semi-definite. For a fixed $a,b,c \in \mathbb R$, what is the best way to determine some $m,n$ which make $M$ positive semi-definite? The eigenvalues of this matrix are $$\lambda=1 \pm (m+n) \pm \sqrt{c^2+m^2+n^2+2cm-2cn-2mn+2t^2}$$

Answer 1

You want the least eigenvalue to be nonnegative. In the case $m+n \ge 0$, this means $m+n <= 1$ and $(1-m-n)^2 \ge c^2 + m^2 + n^2 + 2 c m - 2 c n - 2 m n + 2 t^2$. With $t = a + (m+n)b$, this becomes $$ -2\,{b}^{2}{m}^{2}+ \left( -4\,{b}^{2}+4 \right) mn+ \left( -4\,ab-2\, c-2 \right) m-2\,{b}^{2}{n}^{2}+ \left( -4\,ab+2\,c-2 \right) n-2\,{a} ^{2}-{c}^{2}+1 \ge 0 $$ For fixed $a,b,c$, the boundary of this region in the $m,n$ plane is a conic.