Matrix Similarity

Definition

A first square matrix is similar to a second square matrix if there exists a nonsingular matrix such that the first matrix is identical to the product of the inverse of the nonsingular matrix the second square matrix and the nonsingular matrix.

S Notation

Let $A, B \in \R ^{n \times n}$. $B$ is similar to $A$ if there exists a nonsingular matrix $S \in \R ^{n \times n}$ such that

\[ B = S^{-1}AS. \]

Equivalence Relation