Similar matrices

Definition

Let $F$ be a field and let $A$ and $B$ be $n \times n$ matrices over $F$. The matrix $B$ is similar to $A$ when there is an invertible $n \times n$ matrix $P$ over $F$ with $$B = P^{-1} A P.$$

Remarks

Similarity is an equivalence relation on $M_n(F)$. By the change of basis formula, two matrices are similar if and only if they represent the same endomorphism of $F^n$ in two bases. Similar matrices have the same trace, determinant, characteristic polynomial, rank and eigenvalues. The matrices $I_2$ and $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ have the same characteristic polynomial but are not similar.

Explanations

L1 human

Similar matrices describe the same linear map, seen from two different bases. Properties of the map itself, like trace and determinant, are the same for both matrices. A main goal of linear algebra is to find a simple matrix in each similarity class.