Matrix determinant inequality proof without using information theory

Question

Let $A$ be a $k \times n$ orthogonal matrix; i.e., $AA^T = I_{k \times k}$. For $1 \leq j \leq n$, let the squared norm of the $j$-th column of $A$ be denoted by $\alpha_j^2$; i.e., $$\sum_{i=1}^k a_{ij}^2 = \alpha_j^2.$$ Let $\Lambda = \operatorname{Diag}(\lambda_1, \lambda_2, \dotsc, \lambda_n)$ be a positive definite matrix. Define a function $F$ from the space of p.d. diagonal matrices to $\mathbb R$ as follows: $$ F(\Lambda) = \log \lvert A \Lambda A^T\rvert-\sum_{j=1}^n \alpha_j^2 \log \lambda_j.$$

To show: $F(\Lambda) \geq 0$, with equality only when $\Lambda$ is a multiple of identity.

Background: This result is true and follows from an inequality known in information theory as Zamir and Feder's entropy power inequality. To obtain the result, we may apply Zamir and Feder's inequality to Gaussian random variables with covariances $\lambda_i$. (A statement of the inequality may be found here: Rioul - Information-theoretic proofs of entropy power inequalities, Proposition 9, (65c).)

I am interested in knowing if there is an alternate proof of this result that does not rely on entropy inequalities, and uses linear algebraic tools instead.

Answer 1

Here's a more general result.

From Choi's inequality, we know that for a positive linear map $\Phi$ and an operator convex function $f$, we have $\Phi(f(\Lambda)) \ge f (\Phi(\Lambda))$. Let $\Phi(\Lambda)=A\Lambda A^T$, and let $f(t)=-\log t$, then we obtain \begin{align*} -A\log(\Lambda)A^T \ge -\log(A\Lambda A^T)\\ \Leftrightarrow\qquad\log(A\Lambda A^T) \ge A\log(\Lambda)A^T. \end{align*} This operator inequality is stronger than the what we need. Simply take the trace of both sides to conclude $F(\Lambda) \ge 0$ (using the observation that $\log\det(Z)=\text{tr}\log Z$, and since $\Lambda$ is diagonal).

Answer 2

Let $B := A \Lambda^{1/2}$, so that $A\Lambda A^T = BB^T$. Using the Cauchy-Binet formula for the determinant of $BB^T$, we obtain

$$|BB^T| = \sum_{1 \leq i_1 < \dots < i_k \leq n} |B_{i_1 i_2 \dots i_k}|| B_{i_1 i_2 \dots i_k}^T|, $$

where $B_{i_1 i_2 \dots i_k}$ consists of $k$ columns of $B$ corresponding to the indices $i_1, \dots, i_k$. The right hand side of the above equality may be written explicitly as

$$ \sum_{1 \leq i_1 < \dots < i_k \leq n} \left({\prod_{j=1}^k \lambda_{i_j}}\right) |A_{i_1i_2\dots i_k}|^2.$$

Noting that $\sum_{1 \leq i_1 < \dots < i_k \leq n}|A_{i_1i_2\dots i_k}|^2 = |AA^T| = |I|= 1 $ (again via Cauchy-Binet formula), we can take logarithms and use Jensen's inequality to obtain

$$ \log |A \Lambda A^T| \geq \sum_{1 \leq i_1 < \dots < i_k \leq n} |A_{i_1i_2\dots i_k}|^2 \log \left({\prod_{j=1}^k \lambda_{i_j}}\right). $$

We now gather the coefficients of $\log {\lambda_j}$ for a fixed $j$. Without loss of generality, let $j=1$. The coefficient of $\log {\lambda_1}$ is given by

$$ \sum_{1 = i_1 < \dots < i_k \leq n} |A_{i_1i_2\dots i_k}|^2= 1 - |A_{2,3,\dots, n}A_{2,3,\dots,n}^T| = 1 - |I - A_1 A_1^T| = \alpha_1^2.$$

The first equality follows by using the Cauchy-Binet formula again, the second equality follows from the orthogonality of $A$, and the third equality is true because $|I-uu^T| = 1- ||u||^2$ for any vector $u$.

Answer 3

This is too long for a comment, and not (yet) a complete answer.

---

In other words, we have to show $e^{F(\Lambda)}=\prod_{j} \lambda_j^{-\alpha_j^2}\det (A\Lambda A^\top)\geq 1,$ or $$\prod_{j} \lambda_j^{\alpha_j^2}\leq\det (A\Lambda A^\top).$$ Indeed, this is trivially true for $k=n$, as then $\det (A\Lambda A^\top)=\prod_j \lambda_j$, and by orthogonality of $A$ one has in this case $\alpha_j=1$.

As well, for $k=1$ one has $\det (A\Lambda A^\top)=\sum_j \lambda_j\alpha_j^2\geq \prod_{j} \lambda_j^{\alpha_j^2}$, and this is a consequence of Jensen inequality for the convex function $\exp$. Indeed, the latter gives $$\exp\left(\frac{\sum_j x_j\alpha_j^2}{\sum_j \alpha_j^2}\right)\leq \frac{\sum_j \exp(x_j)\alpha_j^2}{\sum_j \alpha_j^2},$$ and the claim follows by setting $\lambda_j=\exp(x_j)$ and noting that $\sum_j \alpha_j^2=1$.