# Maximum volume submatrices of a Khatri-Rao product of matrix exponentials

My question requires quite a bit of setup, which leads to a conjecture. So I split my question into three parts, Setup, Conjecture, and Question.

Setup:

Pick any two right stochastic matrices $$\boldsymbol{R}_1,\boldsymbol{R}_2 \in \mathbb{R}^{n\times n}$$ and any two positive real numbers $$\mu_1,\mu_2$$ and define two new matrices $$\boldsymbol{Q}_1 := \boldsymbol{R}_1-\boldsymbol{I} \ \ \ \text{and}\ \ \ \boldsymbol{Q}_2 := \boldsymbol{R}_2-\boldsymbol{I},$$ where $$\boldsymbol{I}$$ denotes the standard $$n\times n$$ identity matrix. Next, define two more right stochastic matrices via the matrix exponential $$\boldsymbol{P}_1 := e^{\mu_1\boldsymbol{Q}_1} \ \ \ \text{and}\ \ \ \boldsymbol{Q}_2 := e^{\mu_2\boldsymbol{Q}_2}. \ \ \ \ \ (\ast)$$ Finally define a matrix $$\boldsymbol{P}\in \mathbb{R}^{n^2\times n}$$ entry-wise by $$\boldsymbol{P} (i_1,i_2|j) = \boldsymbol{P}_1(i_1,j)\boldsymbol{P}_2(i_2,j).$$ The rows are denoted by the long indices $$(i_1,i_2)\in \{1,\ldots,n\}\times\{1,\ldots,n\}=:\mathcal{I}$$ (in the usual reverse lexicographical ordering) and the columns by the $$j$$. The matrix $$\boldsymbol{P}$$ can then be represented by the column-wise Khatri-Rao product $$\boldsymbol{P} = \boldsymbol{P}_2\circ\boldsymbol{P}_1.$$ Now pick any subset of rows $$\mathcal{K}= \{(i_1^k,i_2^k)\}_{k=1}^n\subset\mathcal{I}$$ then the submatrix $$\widehat{\boldsymbol{P}}_\mathcal{K}$$ of $$\boldsymbol{P}$$ defined by $$\mathcal{K}$$ has entries given by $$\widehat{\boldsymbol{P}}_\mathcal{K}(k,j) := \boldsymbol{P}_1(i_1^k,j)\boldsymbol{P}_2(i_2^k,j).\ \ \ \ \ \ \ \ \ (\ast\ast)$$ Here we have kept $$\mathcal{K}$$ in reverse lexicographical order and we use $$k$$ to index its members. The entrywise definition $$(\ast\ast)$$ shows that $$\widehat{\boldsymbol{P}}_\mathcal{K} = \widehat{\boldsymbol{P}}_{1,\mathcal{K}}\ast\widehat{\boldsymbol{P}}_{2,\mathcal{K}}$$ where $$\ast$$ denotes the Hadamard Product of matrices and we defined $$\widehat{\boldsymbol{P}}_{1,\mathcal{K}}(k,j) := \boldsymbol{P}_1(i_1^k,j) \ \ \ \text{and } \ \ \ \widehat{\boldsymbol{P}}_{2,\mathcal{K}}(k,j) := \boldsymbol{P}_2(i_2^k,j)$$ for all $$k$$ and all $$j$$.

Conjecture

The subset of rows $$\mathcal{K}= \{(i_1^k,i_2^k)\}_{k=1}^n\subset\mathcal{I}$$ that maximises the volume (absolute value of the determinant) of $$\boldsymbol{P}_\mathcal{K}$$ is always given by $$\mathcal{K} = \{(1,1),(2,2),\ldots,(n,n)\},$$ in which case $$\widehat{\boldsymbol{P}}_{1,\mathcal{K}} = \boldsymbol{P}_1 \ \ \ \text{and } \ \ \ \widehat{\boldsymbol{P}}_{2,\mathcal{K}} = \boldsymbol{P}_2.$$

Question

I came up with this conjecture when I noticed that by generating random matrices of the form described in $$(\ast)$$, counter examples to the conjecture never occurred. I have a hunch that this conjecture is true, but as of yet I have not been able to come up with a proof.

I suspect that the exponential form of $$\boldsymbol{P}_1$$ and $$\boldsymbol{P}_2$$ is important, since I have been able to find general stochastic matrices that are counter examples to the conjecture.

So my question is: Does anyone know results relating to the Khatri-Rao product or matrix exponentials that could be helpful? Or do you know of an approach to proving this conjecture that may be successful?

Cheers in advance for the help!