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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9
votes
2answers
708 views

Determinant of a $k \times k$ block matrix

Consider the $k \times k$ block matrix: $$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & ...
1
vote
3answers
744 views

Detecting if a polynomial is a Pfaffian

Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries? The pfaffian can be defined as $\sqrt{{\rm det}(A) } $ when $A$ ...
0
votes
1answer
94 views

Equivalent determinantal hypersurfaces

I have two matrices $A$ and $B$ (of the same order) whose entries are homogeneous polynomial of the same degree. I have that $\det A=0$ and $\det B=0$ define the same hypersurface of $\mathbb{P}^n$ ...
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} & ...
10
votes
2answers
418 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 $n\times n$ ...
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 ...
3
votes
1answer
187 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} ...
14
votes
4answers
477 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 ...
10
votes
1answer
422 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 ...
8
votes
0answers
226 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 ...
3
votes
0answers
161 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
91 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 $(\cdot)>0$. ...
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} ...
15
votes
2answers
3k 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 ...
6
votes
1answer
266 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 ...
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 ...
28
votes
2answers
653 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 ...
-1
votes
1answer
5k views

Derivative of log determinant and inverse

I have a matrix $\Sigma$ with element $(i,j)$ $$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$ The matrix is positive definite and symmetric (it is a covariance matrix). Now I need to evaluate ...
1
vote
1answer
95 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} ...
2
votes
0answers
105 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 $K$, ...
6
votes
1answer
194 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 $\{-1,1\}$ ...
2
votes
1answer
193 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 ...
5
votes
0answers
87 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
31 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
171 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 ...
9
votes
0answers
104 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 $n$. ...
60
votes
19answers
23k views

Why were matrix determinants once such a big deal?

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
2
votes
1answer
86 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
191 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 ...
5
votes
1answer
1k views

do you know this determinant (basic commutative algebra)?

Let $\ell_1,\dots,\ell_n$ be $d+1$-variate linear forms over complex numbers in variables $X=(X_0,\dots,X_d)$. Consider the $(n-d)$-fold products ...
18
votes
2answers
773 views

a determinantal identity

Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity $$ \det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB) $$ ...
7
votes
1answer
272 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 ...
7
votes
2answers
657 views

When is the determinant a Morse function?

This might be ridiculously obvious, but... For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant ...
3
votes
1answer
130 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) =: ...
26
votes
4answers
1k views

What are the applications of immanants?

Definitions of determinant: $\det(A) = \sum_{\sigma \in S_n}(-1)^{\operatorname{sgn} \sigma}\prod_{i}a_{i, \sigma(i)}$ and permanent: $\mathrm{per}(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, ...
5
votes
0answers
371 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 ...
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
534 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 ...
3
votes
1answer
157 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} ...
20
votes
6answers
3k views

Expressing adj(A) as a polynomial in A?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n ...
3
votes
0answers
925 views

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 ...
4
votes
2answers
783 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} ...
1
vote
5answers
307 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 $B - vv^T$ ...
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 ...
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
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
83 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 ...