Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix

2010-09-18T22:13:31Z

From a physics problem, I am trying to evaluate exactly the following kind of determinant:

G = A + M + N.

A is diagonal M is a product of a column (of 1s) and a row matrix N is a Hermitian Toeplitz matrix.

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).

For completeness, here's the full expression.

$A(m,n) = (m+i\alpha)\delta(m,n)$, $M(m,n) = \beta f(n+\alpha)$, $N(m,n) = -\beta f(m-n)$

$\alpha$ and $\beta$ are real constants. $i$ is $\sqrt{-1}$. $f(x) = (e^{i x t}-1)/x$, and $t > 0$.

Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix - MathOverflow

Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix