What conditions( on $a$ and $b$) I need to impose on the following matrix to make it positive semidefinite?
$$A=\begin{pmatrix}a&b\\b&0\end{pmatrix}.$$
Thanks in advance.
Asked
Active
Viewed 32 times
1 Answers
2
By Sylvester's criterion, $A$ is positive semidefinite if and only if all its principal minors have nonnegative determinant, i.e., if $a\ge 0$ and $\det(A)=-b^2\ge 0$, which means $b=0$.
Dietrich Burde
- 130,978
-
Just to point it out, this also follows from the observation that we need $ax_1^2+2b x_1 x_2\geq 0$, as Neon observed (consider the case $x_1=0,x_2=-b$). – Jonas Dahlbæk Aug 13 '14 at 18:58