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Let $A$, $B$ be $d$ by $d$ matrices where $A$, $A+B$ are symmetric positive semi-definite matrices and $\text{det}B=0$. $f(t)=(\text{det}(A+tB))^{\frac{1}{d}}$ is my function where $t\in[0,1]$.

If $f(t)\leq f(0)+tf'(0)$, then $f$ is concave over $[0,1]$.

I just asked To be concave on $[0,1]$, $f(t)\leq f(0)+tf'(0)$ is enough? for general function, but turns out there is a counterexample.

$f$ is special here, which is a composition of exponent function and determinent, it could be true. Thanks in advance.

My trial : $f(t)=(\text{det}(A+tB))^{\frac{1}{d}}$ looks like a exponent $\frac{1}{d}$ of a polynomial function of order $d$. then $f(t)=(a_{1}t^{d}+\cdots)^{\frac{1}{d}}$.

Second trial : Let $f(t)=(a_{d}t^{d}+\cdots+a_{0})^{1/d}$.

$tf(1/t)=(\det(tA+B))^{1/d}$, LHS$=(a_{d}+\cdots+a_{0}t^{d})^{1/d}$ take $t=0$, then $a_{d}=\text{det}B=0$.

So $f(t)=(a_{d-1}t^{d-1}+\cdots+a_{0})^{1/d}$

We can see $f$ is smooth. Let's claim $f$ doesn't have zero on $(0,1)$.

Suppose not, i.e. $f(z)=0$ for some $z\in(0,1)$. We know $f(1)=\left(\text{det}(A+B)\right)^{1/d}\geq0$.

Then, we can change $A$ into $A+zB$ then the condition $\tilde{f}(t)\leq\tilde{f}(0)+t\tilde{f}'(0)$ doesn't make sence. So $f$ doesn't have any zero on $(0,1)$.

Here, I can't go farther.

kayak
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  • you need special matrices $A,B$ if you want the determinant to be non-negative or positive – Calvin Khor Sep 06 '18 at 08:45
  • @CalvinKhor Thanks for comment, I edited my question. – kayak Sep 06 '18 at 08:48
  • Can you provide more context, and the textbook? also either the claim that $f(0) \in \mathbb R^+$ is false, or in fact A is more than pos. semi-def(or I don't like your notation $\mathbb R^+$). – Calvin Khor Sep 07 '18 at 18:29
  • What do you mean by changing $A$ into $A+zB$, and how did you use the inequality? also, can you tell us what textbook you are using? – Calvin Khor Sep 09 '18 at 13:31
  • @CalvinKhor It's not from a textbook but from an article. and I think my second trial in using $A+zB$ doesn't make sense. – kayak Sep 09 '18 at 13:33
  • Well that's fine, can you share the article? – Calvin Khor Sep 09 '18 at 13:34
  • @CalvinKhor It's called 'Divergence-free positive symmetric tensors and fluid dynamics' by D. Serre. I don't know what's the best way to share with. You can see it page 7 Proposition 1.2 – kayak Sep 09 '18 at 13:35
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    found it on the arxiv https://arxiv.org/abs/1705.00331 – Calvin Khor Sep 09 '18 at 13:40
  • As you said, I don't like my notation $\mathbb{R}^{+}$ lol and you can see that $f$ has an exponent $d-1$ not $d$ but nothing much changed. – kayak Sep 09 '18 at 14:09

1 Answers1

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Some headway... Let $C(t) = A+tB$ for $t\in[0,1]$. Note that $C(t)$ is a convex combination of positive semi-definite matrices $A$ and $A+B$, so it is positive semi-definite. Thus, $\det C(t) \ge 0$ and thus $f(t)$ is well defined for every $t\in[0,1]$.

One can attempt to employ the highschool second derivative test. Define $s := 1/d$, and $g(t):= \det C(t)$ for notational brevity. Then we have

$$ f'(t) = \frac{d}{dt} (g(t)^s ) =sg(t)^{s-1} g'(t)$$ and $$ f''(t) = s(s-1) g(t)^{s-2} g'(t)^2 + sg(t)^{s-1} g''(t)$$

Now observe that

so \begin{align}f''(t) &= s(s-1)g(t)^s [\operatorname{tr}(C^{-1} B)]^2 + s g(t)^s\left([\operatorname{tr}(C^{-1} B)]^2 +0-\operatorname{tr}(C^{-1}BC^{-1}B) \right) \\ &= sg(t)^s \left( s [\operatorname{tr}(C^{-1} B)]^2 -\operatorname{tr}((C^{-1}B)^2) \right) \end{align}

Similarly to the linked question, it seems that the given inequality $ f(t) \le f(0) + tf'(0)$, i.e., $$g(t)^s \le g(0)^s + tsg(0)^{s}\operatorname{tr}(A^{-1}B)$$ is not helpful. It seems like we need to know an inequality between traces of $D:=C^{-1}B$ and $D^2$. As the trace is a sum of eigenvalues, it may be useful to note that
$$ s(\operatorname{tr} D)^2 - \operatorname{tr}(D^2) = s(\sum_i \lambda_i )^2 - (\sum_i \lambda_i^2) $$

If one assumes that $A,A+B$ are better behaved, namely positive definite, then $ f > 0$ is smooth on $(0,1)$, so the above has a chance of working. If in addition the dimension is 2, then there is the equality $$ \det M = \frac12((\operatorname{tr}M)^2 - \operatorname{tr}(M^2) ) $$ so we can notice that $$ f''(t) = \frac12g(t)^{1/2} \left( -\frac12\operatorname{tr}((C^{-1}B)^2)\right) < 0$$ which proves the result true in this special case, without using the inequality $f(t) \le f(0) + tf'(0)$. There are also similar inequalities in dimensions $>2$ but I don't know how to use them.

Calvin Khor
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