**12**

votes

**2**answers

981 views

### Jacobi's equality between complementary minors of inverse matrices

What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...

**2**

votes

**1**answer

236 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} & B_{2n}...

**0**

votes

**0**answers

90 views

### Checking whether the determinant of a matrix is non-zero [on hold]

For a positive integer $n$, let $c$ be the number of ordered integer tripartitions $(a,b,n-a-b)$ of $n$. Let $F = ( f_{ij} )$ be a $c \times c$ matrix where $f_{ij} = (i+1)^{a + b} (i+2)^{b}$ where $i$...

**0**

votes

**0**answers

55 views

### Checking whether a given matrix has a non-zero determinant

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$.
Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is
$M[i,j]={(...

**2**

votes

**0**answers

56 views

### How to find a closed form of following matrix's determinant [closed]

I wanna find a closed form of determinant of the following matrix
$$A(n) =
\begin{pmatrix}
B_{1} & B_{2} & \cdots & B_{n} & 1 \\
B_{n} & B_{1} & \cdots & B_{n-1} &...

**1**

vote

**0**answers

42 views

### Determinant formula related to solutions of a second-order recurrence

Let $A$ be the linear map on the space of complex sequences acting as
$$(Au)_{n}=u_{n-1}+a_{n}u_{n}+u_{n+1}, \quad n\in\mathbb{Z},$$
where $\{a_{n}\}$ is a fixed sequence. Let $f=f(z)$ and $g=g(z)$ be ...

**-3**

votes

**0**answers

59 views

### Minors of a Vandermonde matrix [duplicate]

I am working with the $n$ x $n$ Vandermonde matrix where the "$α_i$'s" form the set of integers from 1 to $n$. That is entry $a_{ij}=i^{j−1}$
What I would like to know is if I delete an equal number ...

**10**

votes

**3**answers

850 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 & \...

**8**

votes

**0**answers

135 views

### Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying
$a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows.
...

**1**

vote

**3**answers

747 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

**1**answer

101 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 $\mathbb{P}^n$ (...

**10**

votes

**2**answers

423 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

**1**answer

98 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

**1**answer

191 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

**4**answers

479 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

**1**answer

432 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 \end{pmatrix}....

**8**

votes

**0**answers

231 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

**0**answers

168 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

**1**answer

92 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

**1**answer

152 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

**2**answers

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

**1**answer

273 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
$$\det(M)\...

**4**

votes

**0**answers

107 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

**1**answer

141 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

**2**answers

668 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

**1**answer

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
$$\frac{\...

**1**

vote

**1**answer

102 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

**0**answers

109 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

**1**answer

196 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

**1**answer

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

**0**answers

90 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 + 3x\...

**0**

votes

**0**answers

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

**1**answer

179 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

**0**answers

106 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

**19**answers

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

**1**answer

90 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

**2**answers

203 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 (1,x,x^2,\cdots,x^{n-...

**5**

votes

**1**answer

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
$$\ell_{i_1}(X)\ell_{i_2}(X)\dots\ell_{i_{n-d}}(X)=\...

**20**

votes

**2**answers

840 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
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...

**8**

votes

**1**answer

304 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

**2**answers

667 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

**1**answer

133 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) =: \textrm{...

**26**

votes

**4**answers

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, \sigma(i)...

**6**

votes

**0**answers

387 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

**1**answer

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

**0**answers

538 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 $s_k:=...

**3**

votes

**1**answer

160 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

**6**answers

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 $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n x^n$....

**3**

votes

**0**answers

936 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

**2**answers

832 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} ...