What is the determinant of a matrix that commutes with a complex structure?

Asked Sep 30 '21 at 13:59

Active Sep 30 '21 at 20:16

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It is known that if $A \in M_{2n}(\mathbb{R})$ is a linear complex structure (i.e. $A^2 = -I$) and $B \in M_{2n}(\mathbb{R})$ satisfies $B^TAB = A$, then $\det(B) = 1$.

But what can we say about $\det(B)$ if it’s invertible and $B^{-1}AB = A$? Is it necessarily positive?

edited Sep 30 '21 at 20:16

asked Sep 30 '21 at 13:59

Pirate Prentice

What is the definition of the complex structure? – blue Sep 30 '21 at 14:55
$A\in M_{2n}(\mathbb{R})$ is a complex structure if $A^2 = -I$. – Pirate Prentice Sep 30 '21 at 15:01
Lukas Heger: I’m asking about the determinant of $B$, not $A$. In any case, this question was already answered. – Pirate Prentice Sep 30 '21 at 20:55

What is the determinant of a matrix that commutes with a complex structure?

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