What are the differences between equivalence, row equivalence and similarity in matrices?

Question

Two matrices being equivalent, row equivalent or similar are introduced in different linear Algebra text books. Each book has a different perspective and different level of difficulty. As a result, I get confused when trying to set apart each quality in terms of the definition, characteristics, ... Also, intuitively, I feel there are mathematical relationships between them. May someone help with this?

Answer 1

1. Similar matrices means that they represent the same linear operator. Two $n\times n$ matrices $A$ and $B$ are similar if there exists a non singular $n\times n$ matrix $P$ such that $$A=P^{-1}BP.$$
2. Equivalents matrices means that they represent the same linear transformation. Two $m\times n$ matrices $E$ and $F$ are equivalent if there exists two non singular matrices $P$ of size $m\times m$ and $Q$ of size $n\times n$ such that: $$E=P^{-1}FQ.$$
3. Two $m\times n$ matrices $M$ and $N$ are row equivalents if one can be obtained from the other using only elementary row operations. This is equivalent to say that there exist a non sigular $m\times m$ matrix $S$ such that: $$M=SN.$$