Smith Normal Form and lower triangular Toeplitz Matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:25:51Z http://mathoverflow.net/feeds/question/32704 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32704/smith-normal-form-and-lower-triangular-toeplitz-matrices Smith Normal Form and lower triangular Toeplitz Matrices Epsilon 2010-07-20T22:10:27Z 2010-09-24T08:02:53Z <p>I am working on a undergrad research project with some other guys. Now the conjecture (unrelated to this question) we are trying to prove boils down to a final subproblem:</p> <p>Let $A (n \times n)$ be a lower triangular Toeplitz matrix (that is, a Toeplitz matrix without the upper triangular part). The Smith invariant factors of $A([k-1,\dots,n],[1,\dots,k-1])$ are also Smith invariant factors of $A([k,\dots,n],[1,\dots,k])$ including multiplicities, where $A([i_1,\dots,i_2],[j_1,\dots,j_2])$ denotes the submatrix of A formed by $i_1$ to $i_2$th rows and $j_1$ to $j_2$th columns and $k\leq [n/2]$; or equivalently, the gcd of all m by m minors of $A([k-1,\dots,n],[1,\dots,k-1])$ equals the gcd of all m by m minors of $A([k,\dots,n],[1,\dots,k])$. $(m\leq k-1)$</p> <p>Since Toeplitz matrix is rather special and this property (though only confirmed by maple) seems quite nice, I am wondering if anyone else has seen this or something similar before? </p> http://mathoverflow.net/questions/32704/smith-normal-form-and-lower-triangular-toeplitz-matrices/39835#39835 Answer by Charles Chen for Smith Normal Form and lower triangular Toeplitz Matrices Charles Chen 2010-09-24T08:02:53Z 2010-09-24T08:02:53Z <p>Some friends (including the original poster!) and I wrote up the proof of the result in <a href="http://arxiv.org/abs/1008.1426" rel="nofollow">this paper</a>. The proof involves symmetric function theory.</p>