Suppose $Z\in S_{++}^n$ (a symmetric positive definite $n\times n$ matrix), $V\in S^n$, $t\in\mathbb R$ such that $Z+tV\in\{X\in S_{++}^n\mid X\preceq 2Y\}$ where $Y\succeq 0$ is given. I need to show that the function:

$$ f(t)=-\log\det Z-\sum_{i=1}^n\log(1+t\lambda_i)-\text{trace}((Z+tV)^{-1}Y) $$

is concave in $t$. The $\lambda_i$ are eigenvalues of $Z^{-1/2}VZ^{-1/2}$. I know that I might be able to show this by showing that $f''(t)\le 0$ but the trace is stopping me from advancing - I don't know what to do with it. Any suggestions?

  • 1
    $\begingroup$ What's "deg"? Do you mean determinant? $\endgroup$ Jan 15 '18 at 8:53
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    $\begingroup$ Also, are $\lambda_i$ the eigenvalues of $V$? $\endgroup$ Jan 15 '18 at 9:01
  • $\begingroup$ @Raskolnikov Made edits $\endgroup$ Jan 15 '18 at 16:44

Hint for the trace term:
1) for fixed $Y$, the trace of $XY$ is linear, with derivative $Y$.
2) the derivative $A^{-1}$ with respect to $t$ is $-A^{-1}\tfrac{dA}{dt}A^{-1}$.
3) Combine 1) and 2) using the chain rule.

  • $\begingroup$ I'm sorry I don't think these are enough hints. So far, using these I have: $$ \frac{d}{dt}\text{trace}((Z+tV)^{-1}Y)=\text{trace}(-(Z+tV)^{-1}V(Z+tV)^{-1}Y) $$ but I'm not sure whether this expression is correct nor where to move forward from it. $\endgroup$ Jan 16 '18 at 2:14
  • $\begingroup$ Good @space_voyager one more derivative. The formulas in Section 2.5.2 of math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf combined with the above should give you the second derivative. Good luck! $\endgroup$
    – max_zorn
    Jan 16 '18 at 3:44
  • $\begingroup$ Thanks, but is the first derivative I wrote correct? In that case the second derivative ($d/dt$) will again just go inside the trace and there is no place to apply the derivative of a trace with respect to a matrix formulae in the Matrix Cookbook. $\endgroup$ Jan 16 '18 at 9:32

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