# Matrix-convexity of inverse of the cofactor matrix

Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ positive semi-definite $\forall \ t \in [0,1]$?)

• I think you meant tf(A) + (1-t)f(B) - f(tA+(1-t)B) positive semi-definite $\forall t \in [0,1]$, right? – Shamisen Feb 13 '15 at 17:55
• Indeed. I corrected that. – Vamsi Feb 13 '15 at 18:08
• @ChristianRemling Why would scaling be a problem? Positive (semi)-definiteness is scale-invariant. – Robert Israel Feb 13 '15 at 18:44
• @Vamsi: this looks like an interesting question. May I ask where it arose? – Suvrit Feb 14 '15 at 20:34

Not just $$3\times 3$$, but in general, the map $$A \mapsto \det(A^{-1})A$$ is operator convex on positive definite matrices.
Proof sketch. $$\newcommand{\pfrac}[2]{\left(\tfrac{#1}{#2}\right)}$$ If suffices to prove the following matrix inequality for two psd matrices $$A, B$$: $$\begin{equation*} \frac{A+B}{\det\pfrac{A+B}{2}} \le \frac{A}{\det{A}} + \frac{B}{\det B}. \end{equation*}$$ Since there exists an invertible matrix $$P$$ such that $$P^*AP=I$$ and $$P^*BP=D$$, where $$D$$ is a positive and diagonal, we may equivalently show that $$\begin{equation*} \frac{I+D}{\det\pfrac{I+D}{2}} \le I + \frac{D}{\det D}. \end{equation*}$$ Proving this inequality reduces to showing $$n$$ inequalities of the form (clearly, it suffices to show one): $$\begin{equation*} \frac{1+d_1}{\prod_{i=1}^n\pfrac{1+d_i}{2}} \le 1 + \frac{d_1}{\prod_{i=1}^n d_i}\quad\equiv\quad 2 \le \pfrac{1+d_2}{2}\cdots\pfrac{1+d_n}{2} + \pfrac{1+d_2}{2d_2}\cdots\pfrac{1+d_n}{2d_n}. \end{equation*}$$ The latter inequality can be shown by induction, so am omitting typing the routine calculations.
End of proof (by Denis Serre). You have to prove $$2^n\le\prod_2^n(1+d_i)+\prod_2^n(1+\frac1{d_i}).$$ Just write the right-hand side as a sum over all elementary monomials (monomials whose partial degrees are $$0$$ or $$1$$): $$\sum_m\left(m(d)+\frac1{m(d)}\right).$$ There are $$2^{n-1}$$ such monomials, and each sum $$m+\frac1m$$ is $$\ge2$$.
Well, this is not an answer. But I cannot resist to mention the following equivalent property. Let $$A\mapsto \hat A$$ denote the cofactor map, and $$B\mapsto \check B$$ its inverse. For positive definite symmetric matrices, $$\hat A=(\det A)A^{-1}$$ and $$\check B=(\det B)^{\frac1{n-1}}B^{-1}$$. Then the harmonic mean is less than or equal to the cofactor mean over the cone of positive definite matrices: $$\langle A^{-1}\rangle^{-1}\le\widehat{\langle\hat A\rangle}.$$