Definition:Matrix Equivalence
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Definition
Let $R$ be a ring with unity.
Let $\mathbf A, \mathbf B$ be $m \times n$ matrices over $R$.
Definition 1
Let there exist:
- an invertible square matrix $\mathbf P$ of order $n$ over $R$
- an invertible square matrix $\mathbf Q$ of order $m$ over $R$
such that:
- $\mathbf B = \mathbf Q^{-1} \mathbf A \mathbf P$
Then $\mathbf A$ and $\mathbf B$ are equivalent.
Definition 2
$\mathbf A$ and $\mathbf B$ are equivalent if and only if they are the relative matrices, to (possibly) different ordered bases, of the same linear transformation.
We write:
- $\mathbf A \equiv \mathbf B$
Also see
- Results about matrix equivalence can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equivalent matrix
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equivalent matrix