Similarity transformation §

Definition For a nonsingular matrix $X$, the map $A\mapsto X^{-1}AX$ is called a similarity transformation of $A$.

Theorem A similarity transformation preserves eigenvalues. Proof We can show this by proving that $A$ and $X^{-1}AX$ share the same characteristic polynomial:

$$p_{X^{-1}AX}(\lambda )=\det (X^{-1}AX-\lambda I)=\det (X^{-1}(AX-\lambda XI))$$

we use Binet theorem and the fact that $XI=IX$ (the identity commutes with every matrix) to get:

$$\det X^{-1}\det (AX-\lambda IX)=\det X^{-1}\det (A-\lambda I)\det X=\det (A-\lambda I)$$

where we again used binet. The last term is the characteristic polynimial of $A$. $\square$