Conditions for positivity of this symmetric real matrix

Question

I have the following real symmetric matrix $M$ of size 3:

\begin{align} M = \begin{pmatrix} a & b & c \\ b & d & e \\ c & e & f \end{pmatrix}, \end{align}

for real parameters $a$, $b$, $c$, $d$, $e$, $f$. What are the necessary and sufficient conditions on these parameters for $M$ to be a positive matrix ($M \geq 0$)?

Answer 1

Since you want to matrix to be positive-semidefinite, the condition is: the coefficients of the characteristic polynomial are alternating. This is equivalent to: for every $k$, the sum of principal $k\times k$ minors is $\ge 0$. In this case, $k$ runs from $1$ to $3$ and we get $$a + d + f \ge 0\\ a d + a f + d f- b^2 - c^2 - e^2 \ge 0\\ a d f + 2 b c e - a e^2 - b^2 f - c^2 d \ge 0$$

An equivalent set of conditions is: every principal minor is $\ge 0$ (apparently stronger than the above, but in fact equivalent).

Note: Leading minors $\ge 0$ may not be enough, as a diagonal matrix starting with $0$ shows.

Answer 2

$D_k : k$-th order principal minor of $A$

$A_{n×n}$ is positive matrix if $D_k\ge 0\space \forall k=1, 2,...,n$

$D_1 : det[a]_{1×1}= a\ge 0$

$D_2 : det \begin{align} \begin{pmatrix} a & b \\ b & d \\ \end{pmatrix} \end{align}=ad-b^2\ge 0$

$\begin{align} D_3&= det \begin{pmatrix} a & b & c \\ b & d & e \\ c & e & f\\ \end{pmatrix} \\ &= a d f-a e^2+b^2 (-f)+2 b c e-c^2 d \ge 0\end{align}$