(reference-request) Theorem of Frobenius regarding transforming integer-valued matrix in diagonal matrix with divisibility conditions A contest paper cites the following theorem as a theorem of Frobenius:
Let $A$ be a $m \times n$ integer-valued matrix. There exists a positive integer $r \leq \min(m,n)$ and two unimodular matrices $P,Q$ such that 
$$PAQ = \text{diag}(d_1,d_2,...,d_r,0,0,...,0),$$
where each $d_i$ divides $d_{i+1}.$
Where can I read more about this result? I'm looking for a proof, and eventually some context.
 A: I have not encountered this theorem before, but with some searching I found these links:
First this, which explains the history of the theorem


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*https://books.google.co.uk/books?id=UzTBBAAAQBAJ&pg=PA269&lpg=PA269&dq=frobenius+theorem+for+matrices+diag+divides&source=bl&ots=sqaMcnBNWJ&sig=ACfU3U0fZ86Y6jxz8NMnOT2V3AkWvSncQA&hl=en&sa=X&ved=2ahUKEwito_rQ4pvhAhW1URUIHXAOCnc4ChDoATACegQICBAB#v=onepage&q=frobenius%20theorem%20for%20matrices%20diag%20divides&f=false
and then this which gives information about the theorem (such matrices are in Smith Normal Form):


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*https://en.wikipedia.org/wiki/Smith_normal_form
and here are some proofs of it:


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*https://mathwanderer.wordpress.com/2009/07/28/smith-normal-form/

*http://jupiter.math.nctu.edu.tw/~weng/courses/adv_alg_2008/4_6_SNF.pdf
Here is a paper with a bunch of theoerms leading up to the construction of Smith normal form:
http://www.numbertheory.org/courses/MP274/smith.pdf
and here is Smith's actual paper presenting his ideas (quite complex and hard to read) regarding this normal form:
https://www.jstor.org/stable/108738
