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.


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 $j$.


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 $$ \widehat{\boldsymbol{P}}_{1,\mathcal{K}} = \boldsymbol{P}_1 \ \ \ \text{and } \ \ \ \widehat{\boldsymbol{P}}_{2,\mathcal{K}} = \boldsymbol{P}_2. $$


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!



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