Similarity of Matrix

Similar

Definition 1. Let $A, B \in M_{n \times n}(F)$. We say that $B$ is similar to $A$ if $\exists Q \in M_{n \times n}$ such that $Q$ is invertible and $B = Q^{-1}AQ$.

Property

Property. Let $A, B \in M_{n \times n}(F)$ be the similar matrices. Then
(a) $A$ and $B$ have the same characteristic polynomial.

Proof. (a) Since $A$ and $B$ are similar, $\exists$ invertible $Q \in M_{n \times n}(F)$ such that $B = Q^{-1}AQ$. Then $f_B(t) = \det(B - tI) = \det(Q^{-1}AQ - tI)$ = $\det(Q^{-1}(A - tI)Q)$ = $\det(Q^{-1})\det(A-tI)\det(Q)$ = $\det(A - tI) = f_A(t)$. $\blacksquare$