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I have two $3\times 3$ matrices each of rank 2.

  1. How can I check that they are equivalent?

  2. What definition of equivalence is there in this case?

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    @Dylan, that seems quite good for equivalence. Or I might go with Sasha's or Michael's approach. I'll try them out and see which one I like (Yes, it's that flaky of a definition...).2011-08-11

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Here is an answer for three conceivable notions of "equivalence":

  1. To find out if the matrices have the same row (column) space, compare the reduced row (column) echelon forms.

  2. 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.

  3. 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.