Two matrices can look very different but represent the same underlying linear transformation, just viewed from different bases.

Equivalence

Two matrices $A$ and $B$ are Equivalent if $B = QAP$, where $P$ and $Q$ are invertible matrices.

• Property: Equivalent matrices have the same Rank .

Similarity

Two square matrices $A$ and $B$ are Similar if $B = P^{-1}AP$.

• Note: Similarity is stricter than equivalence. It’s equivalence where $Q = P^{-1}$.