Let $V=M^{C}_{n \times n}$ with the standard inner product.
Suppose $T_{P}$ is a linear transformation from $V$ to $V$ such as: $\forall X \in V,\ T_{P}X=P^{-1}XP$ for some invertible matrix $P$.Show that the following holds: $(T_{P})^{*}=T_{P^{*}}$ (that is, the conjugate transpose of T equals to T with P conjugate transposed).
In practice I need to show that $(T^{*}_{P})X=(P^*)^{-1}XP^*$, since $(P^{-1})^{*}=(P^{*})^{-1}$.
The examples I've seen so far showed that to find the conjugate transpose of a linear transformation, one needs to find the transformation matrix with respect to some orthonormal basis, and then use the relation $[T^*]_{B}=[T]_{B}^*$. But I somehow doubt that that needs to be done here.
- Are there any properties of the conjugate transpose of a linear transformation that can help me here?
- The linear transformation itself reminds me of matrix similarity, does it have any meaning?