Equivalent matrices

Two matrices ${\displaystyle \mathbf {A} }$ and ${\displaystyle \mathbf {B} }$ are called equivalent matrices (shown as ${\displaystyle \mathbf {A} \sim \mathbf {B} }$) if one can be derived by the other using only left or right multiplication by elementary matrices (p134 in [1]). In other words

${\displaystyle \mathbf {A} \sim \mathbf {B} \iff \mathbf {B=PAQ} }$ for nonsingular ${\displaystyle \mathbf {P} }$ and ${\displaystyle \mathbf {Q} }$.

Row and column equivalence

Two matrices are called

• row equivalent if and only if ${\displaystyle \mathbf {B} =\mathbf {PA} }$  for some nonsingular ${\displaystyle \mathbf {P} }$ , and
• column equivalent if and only if ${\displaystyle \mathbf {B} =\mathbf {AQ} }$  for nonsingular ${\displaystyle \mathbf {Q} }$ .

Interesting note

Perhaps somewhat non-intuitively, two matrices can be equivalent but NOT row or column equivalent. In other words, row or column equivalence are stricter conditions than equivalence. This becomes more explicit when we carefully look at the identities above; e.g., row equivalence implies that we should be able to recover ${\displaystyle \mathbf {B} }$  from ${\displaystyle \mathbf {A} }$  without multiplying by any matrix (other than identity) from right. (See also Exerciose 3.9.2 in [1].)

References

1. Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra"