11
$\begingroup$

A simple question:

If $A$ is a positive semidefinite matrix ($A\succeq 0)$, does it imply that $\mbox{Tr}(A)\geq 0$, where the $\mbox{Tr}(\cdot)$ denotes the trace.

If not, any counter-example? Thanks.

$\endgroup$
  • 1
    $\begingroup$ Suppose a matrix has a negative diagonal element. Can you see that the matrix cannot be positive semidefinite? $\endgroup$ – Daniel Fischer Sep 11 '14 at 19:03
  • $\begingroup$ I know most likely this statement is wrong, but I just could not figure out an counter-example. $\endgroup$ – MIMIGA Sep 11 '14 at 19:06
  • $\begingroup$ @MIMIGA the statement is correct: if $A \succeq 0$, then $\mathrm{Tr}(A) \geq 0$. $\endgroup$ – Omnomnomnom Sep 11 '14 at 19:12
  • $\begingroup$ @Omnomnomnom Is the converse true? This is, if $\mbox{Tr}(A)\geq 0$, then $A\succeq 0$. $\endgroup$ – Diego Fonseca Aug 31 '16 at 15:19
  • 1
    $\begingroup$ @DiegoFonseca No. Consider $$ A = \pmatrix{2&0\\0&-1} $$ $\endgroup$ – Omnomnomnom Aug 31 '16 at 16:11
14
$\begingroup$

Suppose that $A = [a_{ij}]_{i,j=1}^n$ is such that $a_{ii} < 0$ for some $i$. Let $e_i$ be the $i$th standard basis vector; that is, $$ e_i = (\overbrace{0,\cdots,0}^{i-1},1,0,\dots,0) $$ then $e_i^T Ae_i = a_{ii} < 0$, which means that $A$ is not positive semidefinite.

So, if $A$ is positive semidefinite, then all diagonal elements are non-negative, which means that the trace is non-negative.

$\endgroup$
8
$\begingroup$

Yes. If the matrix is semi-positive definite, all the eigenvalues are non-negative. The trace being equal to the sum of eigenvalues, we conclude that the trace has to be non-negative.

$\endgroup$
3
$\begingroup$

We know that positive semi-definite matrices have nonnegative eigenvalues, and the trace of a matrix is equal to the sum of its eigenvalues. Because sum of nonnegative numbers is nonnegative, the trace of a positive semi-definite matrix is nonnegative.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.