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Definition of equivalent matrices

Created over 8 years ago, updated 10 days ago

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:

$$B = PAQ$$

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.

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 $C = (RP)A(SQ)$

Equivalence is coarser than similarity (which requires $Q = P^{-1}$ and only applies to square matrices).