Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.

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3
votes
1answer
185 views

Recursively calculate the determinant

A generic $k \times k$ block symmetric matrix $\Sigma$ is denoted as \begin{align} \Sigma = \begin{bmatrix}\Sigma_{11} & \Sigma_{12} & \ldots & \Sigma_{1k} \\ \Sigma_{21} & \Sigma_{22} ...
10
votes
1answer
416 views

Pfaffian equals complex determinant?

Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure $$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 ...
3
votes
0answers
155 views

Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix $$A = \left(\begin{array}{} 1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots ...
2
votes
1answer
85 views

Inverse Hadamard determinant inequality

As far as I remembered there is an inverse Hadamard inequality for the determinant of the form $$ |D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)} $$ providing all values in $(\cdot)>0$. ...
8
votes
0answers
225 views

Conjecture on matrix with reciprocal principal minors

Some notation: $A(\alpha|\beta)$ is the submatrix of $A \in \mathbb{R}^{n \times n}$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the ...
0
votes
0answers
61 views

A representation problem on graphs

Given adjacency matrix $A\in\{0,1\}^{n\times n}$ of a graph denote $A(y,z)$ to be matrix where $0$ is replaced by $z$ and $1$ by $1-z$ on the non-diagonals (replace diagonals by $y$). Denote the ...
4
votes
1answer
150 views

Real plane cubic curves from points in Gr(3,6) via a certain 6x6 determinant

The following determinant has come up in my research: \begin{align} D(x,y,z)=\det\begin{pmatrix} x & 0 & 0 & \nu_{11} & \nu_{21} & \nu_{31} \\ 0 & y & 0 & \nu_{12} ...
4
votes
0answers
106 views

Hodge duality and the determinant of the product of two matrices

I stumbled onto the following identity, and I would like to know: Is it known by some name and are there some references I might cite (or is it actually too trivial to be mentioned anywhere)? Are ...
5
votes
1answer
137 views

An extension of Hadamard maximum determinant problem

Consider the Vandermonde product $\prod_{1\le j < k \le n} |z_j - z_k|$. It is well-known that under the constraint $|z_j| \le 1$ for all $j$, the product is maximized at a picket fence ...
10
votes
2answers
409 views

How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $ A_i= \biggl(\begin{matrix} C_i+E_i & B_i \\ B_i^T & D_i-F_i \end{matrix} \biggr) $, where $B_i$ is an arbitrary $n\times n$ ...
5
votes
0answers
85 views

Evaluation of Hankel determinants for the reverse Bessel polynomials

Consider the sequence $(\varphi_i)$ of reverse Bessel polynomials which begins as follows. \begin{align*} \varphi_0&=1\\ \varphi_1&=x\\ \varphi_2&=x^2 + x\\ \varphi_3&=x^3 + 3x^2 + ...
0
votes
0answers
30 views

A class of unimodular parametrization

Is there a parametrization of set of matrices $\mathcal M\subseteq\Bbb Z[x_1,\dots,x_{m}]^{n\times n}$ such that $\forall f:\{-1,+1\}^{m}\rightarrow\{-1,+1\}$ $\exists M\in\mathcal M$ such that ...
3
votes
1answer
169 views

On a determinantal equality

In my study, I come across the following curious equality, which I do not know a proof yet (so I am asking it here). Let $k$, $l\in \Bbb Z$ be fixed, $m$ --- the size of the below matrix $M$ --- is ...
6
votes
1answer
192 views

Bounds for maximum determinant of circulant matrices

The Hadamard circulant conjecture states that there do not exist circulant Hadamard matrices with more than $4$ columns. An $n$ by $n$ Hadamard matrix where the entries are chosen from $\{-1,1\}$ ...
9
votes
0answers
103 views

Cycles of length $2^n - 2$ in the De Bruijn graph

It is well known that the number of (cyclic) De Bruijn sequences is $2^{2^{n-1}-n}$. This number may also be interpreted as the number of cycles of length $2^n$ in the De Bruijn graph of order $n$. ...
2
votes
1answer
85 views

How to characterize singular matrix $X$ that solves det$(X−A)=0$, where $A$ is symmetric positive definite?

Consider real square matrices $X$ and $A$ of same size, where $A$ is known to be symmetric positive definite. I came across the matrix equation $XX^{\top} = AX^{\top}$, which solved for $X$ gives ...
8
votes
2answers
186 views

generalizations of Vandermonde matrix to high dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix the maps $$ f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$ x\longmapsto ...
7
votes
1answer
245 views

Exchange determinant and integral of a matrix-valued function

Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M det(A)$ and the determinant of its ...
3
votes
1answer
129 views

Example for Reciprocal Principal Minors

I'm searching for rather specific counter-example. Some notation: $A(\alpha|\beta)$ is the sub matrix of $A$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: ...
5
votes
0answers
369 views

How the idea of adjugate matrix has been designed? [closed]

I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...
3
votes
1answer
95 views

Reference request for a result regarding density of induced probability measure under a submersion

Let $\pi: M \to N$ be a smooth submersion from a bounded open subset of $\mathbb{R}^m$ onto $ N \subset \mathbb{R}^n$, $m \geq n$. Further, let $M$ be given a probability measure $\mu$. Then the map ...
6
votes
1answer
170 views

Determinant of symmetric Latin square

Let $n=2m$ be an even number. Let us construct $n\times n$ symmetric matrices $S_n$ in the following way. The entries are indeterminates $X_1,\ldots,X_{n-1}$. We choose a $1$-factorization of the ...
8
votes
0answers
530 views

Is this 2x2 determinant sequence positive and increasing?

Let $X_1,X_2,X_3$ be a three discrete (integer and non-negative valued) random variables with local probabilities $a_k:=\mathbb{P}(X_1=k)$, $b_k:=\mathbb{P}(X_2=k)$, $c_k:=\mathbb{P}(X_3=k)$ and ...
2
votes
1answer
205 views

Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form $$\begin{bmatrix} \pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\ A_{2n} & ...
4
votes
2answers
738 views

Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case $A = \begin{pmatrix} ...
0
votes
0answers
59 views

When is the discrete logarithmic energy not approximable by its ostensibly more general counterparts?

In my answer at Maximum of the Vandermonde determinant / minimum of the logarithmic energy it is shown that that for each large enough natural $n$ there is some $a=(a_0,\dots,a_{n-1})$ with ...
14
votes
4answers
474 views

Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant $$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i) $$ over all $a_0,\dots,a_{n-1}$ such ...
1
vote
1answer
94 views

Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation

Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of: $$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} ...
3
votes
1answer
155 views

Determinant of a Certain Positive-Definite Block Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
2
votes
0answers
128 views

Determinant Evaluation

Is there a closed form (something involving a ratio of products) for: $$\det\left[\binom{a_i+c}{a_i-i+j}\right]_{1\leq i,j\leq t},$$ where $a_i,c$ are positive integers? I think with $c=0$ this is ...
5
votes
0answers
82 views

Determinantal formulae for Tutte polynomial

Let $G$ be a connected undirected graph. Then the number $ST(G)$ of spanning trees in $G$ equals the following specific value of the Tutte polynomial of $G$: $ST(G)=T_G(1,1)$. On the other hand, ...
0
votes
0answers
89 views

Partial Vandermonde Circulant Determinant Expression

Consider following partial Vandermonde type, circulant matrix $\begin{bmatrix} x_1 & x_2 & 0 & \dots & 0 & x_n\\ x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\ \vdots ...
0
votes
0answers
84 views

Finding a “special” non singular submatrix

Given a square integer matrix $A \in M_n(Z)$ and two subsets $I, J \subset \{ 1, \ldots, n\}$, we define $A_{I,J}$ as the sub-matrix of $A$ containing the rows (resp. columns) whose index is in $I$ ...
7
votes
1answer
229 views

Determinant of some covariance matrix (Gaussian kernel process)

Let $x_1,\dots,x_p$ be $p$ points in $\mathbb{R}^n$ ($n\geq 2$) with $x_1=0$. Consider the symmetric matrix $M(x)=(m_{ij}(x))_{1\leq i,j\leq p}$ where $m_{ij}(x) = \exp(-\frac{1}{2}\Vert x_i - ...
6
votes
1answer
103 views

Sum identities with immanants

For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show $$ \sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} ...
1
vote
1answer
121 views

inverse Laplace transform of the determinant

what is the inverse Laplace transform of the function $x\mapsto\det x$? (where $x\in\mathbb{R}^{n\times n}$ is a symmetric metric, if necessary at all.)
1
vote
0answers
176 views

Is there a way to simplify this apparently huge characteristic polynomial calculation?

Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let ...
49
votes
7answers
4k views

How to prove this determinant is positive?

Given the matrices $ A_i= \biggl(\begin{matrix} 0 & B_i \\ B_i^T & 0 \end{matrix} \biggr) $, where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove that $\det(I + ...
10
votes
2answers
353 views

Determinant of a checkerboard Hankel matrix with Catalan numbers

My goal is to compute \begin{equation} I = \det \left(\mathbf{I} + \mathbf{A}\right) \end{equation} where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers: $$ \left\{ ...
2
votes
0answers
191 views

An (open?) problem about a sequence of nested principal sub-matrices and their determinants

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
6
votes
1answer
264 views

Geometric interpretation of the Desnanot-Jacobi Identity

Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the i-th row and j-th column of $M$. The Desnanot-Jacobi Identity states ...
2
votes
2answers
213 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
241 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 ...
2
votes
0answers
198 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? ...
2
votes
0answers
104 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 $K$, ...
1
vote
0answers
62 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 ...
3
votes
0answers
80 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 ...
28
votes
2answers
647 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 ...
6
votes
0answers
208 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
160 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 ...