I have two $3\times 3$ matrices each of rank 2.
How can I check that they are equivalent?
What definition of equivalence is there in this case?
I have two $3\times 3$ matrices each of rank 2.
How can I check that they are equivalent?
What definition of equivalence is there in this case?
Here is an answer for three conceivable notions of "equivalence":
To find out if the matrices have the same row (column) space, compare the reduced row (column) echelon forms.
Usually, matrices $A$ and $B$ of the same dimensions are called equivalent if there are invertible matrices $S$ and $T$ such that $A = SBT$. Two matrices of the same dimensions are equivalent iff they have the same rank. So without further computations, your two rank 2 matrices are equivalent.
Another related notion is similarity: Two $n\times n$ square matrices $A,B$ are called similar if there is an invertible $n\times n$ matrix $S$ such that $A = SBS^{-1}$. Two matrices are similar iff they have the same rational normal form. In the case that your base field is algebraically closed, you may also compare the Jordan normal forms.