Definition:Matrix Similarity
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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 $R$.
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.
We write:
- $\mathbf A \sim \mathbf B$
Also see
- Results about matrix similarity can be found here.