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Given the matrices $X$ and $Y$ in $[0,1]^{n\times m}$, for $n > m > 3$, so that $X1_m=1_m$ and $Y1_m=1_m$, where $1_m$ denotes a $m$-length column vector of ones, find a matrix $Q$ in $R^{m\times m}$ …

For matrices $X,Y\in [0,1]^{n\times m}$, for n > m, is there a square matrix $W\in R^{n\times n}$ so that $X^TWY$ is diagonal if and only if $Y = X$? Furthermore, $X$ and $Y$ are column normalized so …

Here is a solution: $W = X(X^TX)^{-1}D(X^TX)^{-1}X^T$ for a diagonal matrix $D$. You may verify that $X^TWY = D$ iff $Y = X$.

Given a pair of distributions $x,y\in(0,1]^{n\times 1}$, so that $1^Tx=1$ and $1^Ty=1$, I want to build a matrix $C$ (change matrix) that satisfy at least the following properties: i) $C$ is diagona …