Eigenvalues and determinant of conjugate, transpose and hermitian of a complex matrix.

For a strictly complex matrix $A$,

1) Can we comment on determinant of $A^{*}$ (conjugate of entries of $A$) , $A^{T}$ (transpose of A) and $A^{H}$ (hermitian of $A$). I know that for real matrices, $\det(A)=\det(A^{T})$. Does it carry over to complex matrices, i.e. does $\det(A)=\det(A^{T})$ in general? I understand $\det(A)=\det(A^{H})$ (from Schur triangularization).

2) The same question as first, now about eigenvalues of $A$. I would like to know about special cases, for instance what if $A$ is hermitian or positive definite and so on.

Solution 1:

Since complex conjugation satisfies $\overline{xy} = \overline{x} \cdot \overline{y}$ and $\overline{x+y} = \overline{x} + \overline{y}$, you can see with the Leibniz formula quickly that $\det[A^*] = \overline{\det[A]}$.

For complex matrices $\det[A] = \det[A^T]$ still holds and doesn't require any changes to the proof for real matrices.

Together this means that $\det[A] = \overline{\det[A^H]}$.

This applies to the eigenvalues as well: the characteristic polynomial of $A^*$ is given by $\det[tI - A^*] = \det[(\overline{t}I - A)^*] = \overline{\det[\overline{t}I - A]}$ and the eigenvalues of $A^*$ are exactly the complex conjugates of those of $A$.

In particular if $A$ is hermitian, $A = A^*$ and so all eigenvalues are equal to their complex conjugates - in other words, they're real.