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$$ [(U-V)^{-1}X^{T}Y]^{T}\;[(U-V)^{-1}X^{T}Y] = I. $$

Here $U$ and $V$ are symmetric $d \times d$ matrices; $X=[x_1,x_2,...,x_n]$ is an $n \times d$ data matrix ($n$ is the number of samples and $d$ is the number of data dimensions); $Y \in \{0,1\}^{n \times c}$ is an label matrix ($c$ is the number of classes). When $x_i$ belongs to class $j$, $1\leq j\leq c$, the $i$-th row and $j$-th column of $Y$ are set to 1.

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  • $\begingroup$ I guess that you cannot get $V$. From your equation, it follows only that $Q=(U-V)^{-1}X^TY$ is orthogonal. But there is a lot of orthogonal matrices. If you know $Q$, then $V=U-X^TYQ^T$. $\endgroup$
    – Janko Bracic
    Commented Nov 17 at 5:49
  • $\begingroup$ This is not an undergraduate assignment, but a part of a research project. Your statement is correct. $Q^TQ=I$ is a constraint condition, and I need to obtain a $V$ that satisfies $Q^TQ=I$. In this equation, only $V$ is unknown. $\endgroup$
    – dawei
    Commented Nov 17 at 7:41
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    $\begingroup$ People here are research mathematicians, we don't need to be told what transpose or inversion is for matrices :-) And the algebraic problem doesn't care about the source of the information (data/class etc) unless these imply specific algebraic properties of the matrices. $\endgroup$
    – David Roberts ♦
    Commented Nov 17 at 10:05
  • $\begingroup$ So, to ask a clarifying question: what does a "label matrix" look like? What, if any, conditions to its entries satisfy? (I took the liberty of rewriting the question to be a bit more concise). $\endgroup$
    – David Roberts ♦
    Commented Nov 17 at 10:09
  • $\begingroup$ @JankoBracic This looks like a good answer, I suggest you post it as one. $\endgroup$
    – Federico Poloni
    Commented Nov 17 at 10:16

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