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I want to find the minimum to the following problem:

$$ \min_{X} \|f(X)\|_F \label{1}\tag{1} $$ where $X$ is a rectangular matrix, and $f$ is a function of it, involving other matrices, the norm is Frobenius norm.

I want to know if the solution of problem \eqref{1} is equivalent to the following problem: $$ \min_{X} \|f(X) A\|_F \tag{2} $$ where $A$ is a rectangular matrix.

Are the two problems equivalent? if not, are there some properties on $A$ so that they become equivalent?

Thank you in advance.

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    $\begingroup$ Equivalent in what sense? $\endgroup$
    – Brendan McKay
    Commented Aug 31, 2022 at 12:32
  • $\begingroup$ @BrendanMcKay in the sense that the minimum is the same, i.e they have the same solution $\endgroup$
    – nazarem
    Commented Aug 31, 2022 at 15:19
  • $\begingroup$ X_{optimal} for problem (1) is the same for problem (2) $\endgroup$
    – nazarem
    Commented Aug 31, 2022 at 15:45
  • $\begingroup$ They are trivially equivalent if $AA^T=I$. $\endgroup$
    – Federico Poloni
    Commented Aug 31, 2022 at 19:12

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