# Definition of equivalent matrices

Put content here**Definition:** Two $m \times n$ matrices $A$ and $B$ are **equivalent** if there exist invertible matrices $P$ ($m \times m$) and $Q$ ($n \times n$) such that:
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$$B = PAQ$$
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**Interpretation:** $P$ represents a change of basis in the codomain and $Q$ represents a change of basis in the domain. Equivalent matrices represent the same linear transformation $T: V \to W$ with respect to different choices of bases.
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**Properties:**
- Reflexive: $A = I \cdot A \cdot I$
- Symmetric: if $B = PAQ$ then $A = P^{-1}BQ^{-1}$
- Transitive: if $B = PAQ$ and $C = RB S$, then $C = (RP)A(SQ)$
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Equivalence is coarser than similarity (which requires $Q = P^{-1}$ and only applies to square matrices).

# Parents

* Matrix equivalence