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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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Element-by-element equivalence I assume. – Phonon Aug 10 '11 at 19:40
Same characteristic polynomials is necessary condition for equivalence. – Sasha Aug 10 '11 at 19:45
@Phonon, if the third row is simply 3 times the second row in one of the matrices, then element-by-element equivalence won't help. – Unapiedra Aug 10 '11 at 19:46
So... you need to check whether they are equivalent, but you don't know the definition of "equivalent"? You'll have a very hard time checking it, then! – Arturo Magidin Aug 10 '11 at 19:55
As an initial guess, I thinking "equivalent" means having the same column space. Just a guess. – Michael Hardy Aug 11 '11 at 4:50
up vote 1 down vote accepted

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.

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