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The transpose suggested in a comment was actually on the wrong B term.
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PThomasCS
PThomasCS
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Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions: $$ B(B^\intercal AB)^\star B = A^\star, $$$$ B(B^\intercal AB)^\star B^\intercal = A^\star, $$ and such that $A^\star=0$ if and only if $A=0$?

Edit: Also, $\star : \operatorname{M}(m,n,\mathbb R) \to \operatorname{M}(n,m,\mathbb R)$.

Edit: If possible, we would also like $\operatorname{rank}(A^\star)=\operatorname{rank}(A)$.

Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions: $$ B(B^\intercal AB)^\star B = A^\star, $$ and such that $A^\star=0$ if and only if $A=0$?

Edit: Also, $\star : \operatorname{M}(m,n,\mathbb R) \to \operatorname{M}(n,m,\mathbb R)$.

Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions: $$ B(B^\intercal AB)^\star B^\intercal = A^\star, $$ and such that $A^\star=0$ if and only if $A=0$?

Edit: Also, $\star : \operatorname{M}(m,n,\mathbb R) \to \operatorname{M}(n,m,\mathbb R)$.

Edit: If possible, we would also like $\operatorname{rank}(A^\star)=\operatorname{rank}(A)$.

Added that the operator flips the dimensions of the matrix.
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PThomasCS
PThomasCS
  • 399
  • 1
  • 9

Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions: $$ B(B^\intercal AB)^\star B = A^\star, $$ and such that $A^\star=0$ if and only if $A=0$?

Edit: Also, $\star : \operatorname{M}(m,n,\mathbb R) \to \operatorname{M}(n,m,\mathbb R)$.

Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions: $$ B(B^\intercal AB)^\star B = A^\star, $$ and such that $A^\star=0$ if and only if $A=0$?

Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions: $$ B(B^\intercal AB)^\star B = A^\star, $$ and such that $A^\star=0$ if and only if $A=0$?

Edit: Also, $\star : \operatorname{M}(m,n,\mathbb R) \to \operatorname{M}(n,m,\mathbb R)$.

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PThomasCS
PThomasCS
  • 399
  • 1
  • 9

Existence of generalized inverse-like operator

Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions: $$ B(B^\intercal AB)^\star B = A^\star, $$ and such that $A^\star=0$ if and only if $A=0$?