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I suspect this is true since it holds in the case over a field. But suppose $A\in M_{m\times n}(R)$ where $R$ is a PID. Does it still hold that $A$ and $A^{T}$ have the same invariant factors?

Regards,

2 Answers 2

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The answer is yes. Indeed, if $d_1,\dots,d_r$ are the invariant factors, and if $d_i$ divides $d_{i+1}$ for $1\le i < r$, then $d_1\cdots d_i$ is the gcd of the $i$-minors of any presentation matrix.

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    Thanks, but why does that imply they have the same invariant factors?2011-10-29
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    Dear @Keith: You're welcome. Do you agree that a matrix and its transpose have the same $i$-minors?2011-10-29
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    I'd never thought of this, but I think I've convinced myself of this fact now.2011-10-29
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If you know that the invariant factors are the entries on the diagonal after diagonalizing the matrix by (perhaps generalized) row and column operations, then, given row and column operations that diagonalize A, ask yourself what the corresponding column and row operations do to the transpose.

The moral I am trying to convey is, that to answer this question, one should begin with a definition or computation of the invariant factors, and then ask what happens when that same characterization is applied to the transposed matrix. I.e. one should ask whether the given construction is in fact unchanged by taking transpose. This describes also M. Gaillard's answer.

Another equivalent point of view prceeds from the knowledge that the invariant factors of A occur on the diagonal of a diagonal matrix D obtained as a product of form XAY = D. Then ask if one can find a similar equation involving the transpose of A.

If you have in fact some other characterization of the invariant factors, I urge you to use that one to give another answer.

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    IMHO this does answer the question (and rather well). But, at the moment it is phrased as a question. This ain't Jeopady :-)2016-12-25
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    So on this site, answering a question is unacceptable if it is done with a question mark? Pardon me but after 40 years as a math teacher I have learned that is usually more helpful to stimulate the questioner to answer his own question, rather than to simply state bluntly the answer. I.e. to me any statement that reveals the answer, does indeed answer the question. But I am not the arbiter of taste here. Thank you for clarifying.2016-12-25
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    I.e. I usually try to engage the questioner in a discussion of his topic and leave it to him/her to decide if it has been answered. That way the questioner must do a bit of thinking rather than being handed an edict to accept. But that's just my opinion. It may be unsuitable. However compare the second comment above to the answer by Pierre-Ives Gaillard who has chosen exactly the same way to complete his answer, i.e. with a question. Mr. Richards seems to have benefited from that question, according to his own following comment.2016-12-25
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    Ok, thanks to those who have enlightened me on the rules of engagment. I have tried to give essentially the same answer without question marks.2016-12-25
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    What I'm not sure is for whom you answered the question in a way which invites to a dialog. As far as I can see the last OP's visit on this site is dated more than five years ago.2016-12-25
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    a retired teacher's hope springs eternal. this also reminds me of the detailed comments i continued to write on hundreds of student exams over three decades, even though essentially no students ever came back to look at them!2016-12-25