2 edited tags
1

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

Hi

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 $t > 0$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 9, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9387175440788269, "perplexity_flag": "head"}