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9
votes
1answer
336 views

Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$. Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$: $$R_q = M_q^T (M_q ...
2
votes
2answers
156 views

Estimating a Selberg-type integral (or a Fredholm determinant)

I am concerned with the asymptotical behavior of integrals like this for large $n$ $$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-x_{j}^{2}}dx_{j},$$ ...
2
votes
2answers
120 views

How to calculate one Cauchy type determinant

As we know, a Cauchy determinant of size n admits the following explicit formula: $$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y ...
1
vote
0answers
142 views

Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group. How does one go about finding that common basis which will find the irreducible components of all of them simultaneously? ...
7
votes
1answer
418 views

Determine if a matrix is unimodular

Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?
1
vote
0answers
40 views

Approximate determinantal point process

Consider a random process defined on $2^{\mathcal{X}}$, i.e. all subsets of a set $\mathcal{X}$. It's well known that this process is determinantal if one can find a positive semidefinite matrix K, ...
13
votes
1answer
891 views

slick-proof-of-trick-for-counting-domino-tilings

The trick for rewriting the number of domino tilings of a simply-connected finite lattice region as the absolute value of the determinant of a matrix (due I believe to Kasteleyn and Percus, but if ...
5
votes
2answers
662 views

Proving that the kernel of this matrix is of dimension 2

(Edit : see at the bottom of the question for an additional surprising possible hint.) Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when ...
1
vote
0answers
44 views

How to define the determinant of a morphism between graded Lie algebras?

I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...
5
votes
1answer
209 views

Vanishing patterns of minors of matrix

Let $M$ be a $m\times n$ matrix with entries in, say $\mathbb{C}$; assume $n\leq m$. Denote by $I\subseteq\{1,2,\ldots, n\}$ a subset of the columns of M. I am interested in positive results to the ...
19
votes
1answer
441 views

How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity To ...
0
votes
2answers
97 views

About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$) I guess that the eigenvalues of $B - vv^T$ ...
3
votes
0answers
62 views

Where does this identity involving sums of Hankel-like determinants over partitions come from?

For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix ...
6
votes
0answers
186 views

Does anyone know this determinant?

The following determinant arises in a combinatorial enumeration problem. I wonder if anyone has seen it before in any context or knows how to evaluate it. I tried computing it for small $n$ but didn't ...
0
votes
1answer
142 views

Are these particular kinds of matrices well known?

Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that, all the diagonal entries are either $a$ or $a+1$ all the non-zero off-diagonal entries are ...
1
vote
1answer
136 views

Are there good ways of relating a minor to the full determinant?

Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
0
votes
2answers
326 views

A sum-of-determinants identity [closed]

I posed a terser version of this question on math.stackexchange.com and after 24 hours I get only a comment on a detail of notation and neither votes nor answers. Suppose ...
1
vote
0answers
107 views

How to see that this pairing of line bundles is multiplicative?

Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one. For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...
3
votes
1answer
168 views

Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries $$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\ -1 ...
6
votes
3answers
388 views

An infinite product associated with random matrices

Motivation Let ${\mathbb F}_q$ be the field with $q$ (a power of some prime number) elements. Then the order of $GL_n({\mathbb F}_q)$ is $$(q^n-1)(q^n-q)\cdots(q^n-q^{n-1}).$$ The fact that this ...
0
votes
1answer
47 views

Determinant of block covariance matrix [closed]

I wonder how to express the determinant of a block covariance matrix. For example, I have a covariance matrix $\Sigma=\left[ \begin{array}{cc} \Sigma_1 & \Sigma_{12} \\ \Sigma_{21} ...
10
votes
3answers
648 views

A class of matrix determinants between Wronskians and Vandermondes

Update: see below Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions ...
4
votes
1answer
229 views

Generalized Cauchy-Binet sum over a fixed subset of indices

I originally posted this on math.stackexchange, but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now. Apologies ...
1
vote
0answers
72 views

Does this permanent have a closed form?

What is the closed form of this permanent? (similar to the Cauchy determinant) \begin{aligned} f(z_1,z_2,\cdots,z_N,w_1,w_2,\cdots,w_N)=\left[ \small{\begin{matrix} \frac{1}{(z_1-w_1)^2} && ...
1
vote
0answers
76 views

MInors related problem [closed]

A matrix $A$ has $m$ rows and $n$ colums, such that $m \leq n$. We know that each row of $A$ has the norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
3
votes
1answer
256 views

Number of Matrices with bounded determinant

Here's my question: Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...
10
votes
2answers
394 views

Are bounds known for the maximum determinant of a (0,1)-matrix of specified size and with a specifed number of 1s?

The problems of determining the maximum determinant of an $n \times n$ $(0,1)$-matrix and the spectral problem of determining exactly which other determinants can possibly occur are both reasonably ...
7
votes
1answer
321 views

Block Matrix determinant

Consider the $k \times k$ block matrix: $ C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \ddots & ...
10
votes
1answer
212 views

Factor a sum of products of cofactors

Let $M$ be any $n\times n$ matrix. We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$. We can write ...
3
votes
3answers
3k views

Derivative of a determinant of a matrix field

Let $A(x_1,...,x_n)$ be an $n\times n$ matrix field over $R^n$. I am interested in the partial derivative determinant of $A$ in respect to $x_i$. In can be shown that: ...
2
votes
0answers
84 views

$G$-invariant part of products of determinants of minors

Let $G = SL_n$; then for any tuple $\lambda$ such that $\sum \lambda_i = n$, define $f_\lambda(g)$ as the product of the determinants of successive minors of lengths $\lambda_i$ of $g$ (e.g. for ...
12
votes
2answers
2k views

Determinants in Graph Theory

In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various properties in themselves. For example, their trace can be calculated (it ...
8
votes
0answers
165 views

Sum over growing Young tableaux

Let $\lambda_0,\lambda_1,\lambda_2,\lambda_3,\ldots$ be a sequence of Young diagrams, such that each successive diagram is obtained from the prior by the addition of one box (don't forget that the row ...
3
votes
4answers
361 views

Determinant of sum of Kronecker products

Given four real symmetric matrices $A,B \in \mathbb{R}^{n \times n}$ and $C,D \in \mathbb{R}^{m \times m}$, is there an efficient way to compute the determinant: $\det|A \otimes C + B \otimes D |$
9
votes
2answers
1k views

When the determinant of a 2x2 polynomial matrix is a square?

Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...
15
votes
3answers
1k views

How to show a certain determinant is non-zero

For any $n$ distinct points $x_1,x_2 , \ldots , x_n$ on the real line show that the matrix $M$ where $M(i,j) = e^{\lambda_j x_i} $ has non-zero determinant where $\lambda_1 \lt \lambda_2 \lt \ldots ...
5
votes
1answer
401 views

Characteristic polynomials of certain random symmetric matrices and the complexity of random Morse functions

Investigations concerning random Morse functions led me to the following problem. Consider the classical GOE of $m\times m$ real symmetric matrices $A$ with independent Gaussian entries with ...
-1
votes
1answer
118 views

Determinant of a sum of two Hankel matrices [closed]

First version: Let $A$ and $B$ be (complex) Hankel matrix. Is it true that $\det (A+B)\neq 0$ if $\det A=0$ and $\det B\neq0$? No. Reformulating: For which $B$ is it true that $\det (A+B)\neq 0$ if ...
9
votes
1answer
339 views

Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle

Let $z_{1},\dots,z_{k}$ be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$ consider the rectangular Vandermonde matrix $$ V_{N}=\begin{pmatrix}1 ...
1
vote
0answers
74 views

range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
3
votes
1answer
256 views

Reconstructing a (unitary) matrix from the determinant of its sub-matrices

I want to find the unitary $N \times N$ matrix U from the following data. Let $M$ be an integer $(1< M<N-1)$ and let $\mathcal S$ be the space of all the possible subsets of $\{1,2,\dots, N\}$ ...
0
votes
1answer
157 views

A determinant problem with symmetric PSD matrices

Suppose we have a a set of matrices in the complex field of the form $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ where $a_i$ are constant positive real scalars and $v_i$ are constant complex valued finite ...
10
votes
2answers
519 views

A binomial determinant fomula

Is there an existing or elementary proof of the determinant identity $ \det_{1\le i,j\le n}\left( \binom{i}{2j}+ \binom{-i}{2j}\right)=1 $?
-2
votes
1answer
342 views

How to obtain the determinant of the difference of two matrices? [closed]

I am trying to obtain the determinant of the difference between the identity matrix and an A matrix. The question is such: ...
2
votes
3answers
313 views

LU decomposition

Consider a $N \times N$ symmetric real matrix $A$: $A_{ij} = (\sum_{k=1}^N n_{ik}) \delta_{ij} - n_{ij}$, where $n_{ij}$ is a real symmetric matrix whose elements are equal to $1$ or $0$. $A$ has one ...
6
votes
1answer
272 views

Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant: Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
1
vote
1answer
126 views

Estimate the determinant of sparse 0-1 matrix

There is a matrix A where each entry is either 0 or 1. Each column has exactly a 1's and each row has at most b 1's. What's the upper bound of abs(|A|)? The condition is stronger than the Hadamard's ...
13
votes
2answers
779 views

How to invert the matrix [n choose 2j - i] ?

In a certain model of a stat-physics type, one encounters a matrix $$ A_n:=\left[\binom{n}{2j-i}\right]_{i,j=1}^{n-1}. $$ The determinant of this matrix (equal to $2^{\binom n2}$) counts the number ...
20
votes
3answers
2k views

The Wronskian of sin(kx) and cos(kx), k=1…n

What is the determinant of the Wronskian of the functions $\{\cos\ x, \sin\ x, \cos\ 2x, \sin\ 2x,\ldots, \cos\ nx, \sin\ nx\}$? This determinant seems to be an integer, and the sequence starts with ...
6
votes
3answers
2k views

Sarrus determinant rule: references, extensions

SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS An undergraduate came to me with an identity for 4x4 determinants that is actually correct: $\det(A)=h(A)+h(RA)+h(R^{2}A)$ where R cyclically ...