Definition:Matrix Similarity/Definition 2

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 $R$.

$\mathbf A$ and $\mathbf B$ are similar if and only if they are the relative matrices, to (possibly) different ordered bases, of the same linear operator.

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