Conjugate linear operators

Linear operators $\alpha$ and $\beta$ on the same finite-dimensional vector space are conjugate when $\beta =s^{-1}\alpha s$ for some linear isomorphism $s$. Their matrices in any fixed basis are similar.

For an invertible operator $\beta$ on $V$, the mapis a linear isomorphism. Conjugate choices of $\beta$ induce conjugate operators ${\varphi }_{\beta }$.