Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:51:31Z http://mathoverflow.net/feeds/question/39255 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39255/determinant-of-a-sum-of-a-diagonal-matrix-a-dyadic-product-matrix-and-a-hermiti Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix dee 2010-09-18T22:13:31Z 2010-09-19T03:02:36Z <p>Hi</p> <p>From a physics problem, I am trying to evaluate exactly the following kind of determinant:</p> <p>G = A + M + N.</p> <p>A is diagonal M is a product of a column (of 1s) and a row matrix N is a Hermitian Toeplitz matrix.</p> <p>It would be of great help to me if anyone could point out known techniques. I've attempted various decompositions and had no luck. Further, I am more interested in the continuum limit of this determinant (i.e. when the matrix size N -> infinity, and the matrix indices are suitably taken to some continuous variable).</p> <p>For completeness, here's the full expression.</p> <p>$A(m,n) = (m+i\alpha)\delta(m,n)$, $M(m,n) = \beta f(n+\alpha)$, $N(m,n) = -\beta f(m-n)$</p> <p>$\alpha$ and $\beta$ are real constants. $i$ is $\sqrt{-1}$. $f(x) = (e^{i x t}-1)/x$, and $t > 0$.</p>