Definition:Matrix Similarity/Definition 1

Definition

Let $R$ be a ring with unity.

Let $n \in \N_{>0}$ be a natural number.

Let $\mathbf A, \mathbf B$ be square matrices of order $n$ over $R$.

Let there exist an invertible square matrix $\mathbf P$ of order $n$ over $R$ such that $\mathbf B = \mathbf P^{-1} \mathbf A \mathbf P$.

Then $\mathbf A$ and $\mathbf B$ are similar.

We write:

$\mathbf A \sim \mathbf B$

Also see

• Equivalence of Definitions of Matrix Similarity

• Results about matrix similarity can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 16$: Equivalence relations (Worked Example)