Equivalence of Definitions of Matrix Similarity

Theorem

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 $R$.

The following definitions of the concept of Matrix Similarity are equivalent:

Definition 1

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.

Definition 2

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

Proof

This is specifically demonstrated in the corollary to Change of Basis Matrix under Linear Transformation.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices