**7**

votes

**1**answer

475 views

### Does this Linear Algebra Construction have a Name?

Let $\mathcal{R}$ be a ring and let $v^0,\ldots,v^{k-1}\in\mathcal{R}^m$ with $m \geq k$. Suppose we wish to find $w\in Span(v^0,\ldots,v^{k-1})$ such that $k-1$ specified coordinates of $w$ vanish ...

**1**

vote

**1**answer

58 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} ...

**8**

votes

**0**answers

173 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 ...

**7**

votes

**0**answers

477 views

### Can one give a “nice” expression for this determinant?

I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague.
...

**6**

votes

**0**answers

198 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 ...

**5**

votes

**0**answers

64 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, ...

**5**

votes

**0**answers

96 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

**0**answers

552 views

### determinant of fibonacci-sum graphs

We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$.
The adjacency matrix of this graph is $A= (a_{ij})$ so that
$a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence;
$a_{ij}=0$ ...

**3**

votes

**0**answers

75 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 ...

**3**

votes

**0**answers

267 views

### det(A)det(B) = det(AB+correction), Capelli identities, “factorzied” representation of gl_n

Context: some probably know that there are Capelli identities which state
det(A)det(B) = det(AB+correction) for some matrices with non-commuting elements, they go back to 19-th century, but also ...

**3**

votes

**0**answers

838 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 ...

**2**

votes

**0**answers

102 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 ...

**2**

votes

**0**answers

180 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 ...

**2**

votes

**0**answers

87 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 ...

**2**

votes

**0**answers

279 views

### Morphisms of Spectral Sequences and alternating products

Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences.
Assume that morphisms ...

**2**

votes

**0**answers

134 views

### which deformation of a matrix lead to flat deformations of determinantal varieties (fitting ideals)?

Let $(R,m)$ be a complete local ring over a field (of char=0). Consider a (not necessarily square) matrix $A$ over $R$. Consider its fitting ideal, $I_j(A)$. In general, a deformation of the matrix, ...

**2**

votes

**0**answers

251 views

### Function recursion relation over symmetric group

Hi!
Let P be a permutation in the symmetric group SN and let π=πj, j+1 be a transposition of elements j and j+1 of the permutation. Let A(P) be a function in dependence of the permutation P. ...

**1**

vote

**0**answers

155 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 ...

**1**

vote

**0**answers

158 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?
...

**1**

vote

**0**answers

62 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

**0**answers

52 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 ...

**1**

vote

**0**answers

112 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 ...

**1**

vote

**0**answers

74 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

**0**answers

83 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 ...

**1**

vote

**0**answers

141 views

### Polynomials satisfying a three-term recurrence

Let ${p_n}(x) = x{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x)$ for some numbers ${a_n}$ with initial values ${p_{ - 1}}(x) = 0$ and ${p_0}(x) = 1.$
By Favard’s theorem about orthogonal polynomials ...

**1**

vote

**0**answers

163 views

### Generalized Schur polynomial from block Toeplitz matrices

By using the Jacobi-Trudi identity, one may interpret banded Toeplitz matrices, and minors of such matrices in terms of Schur polynomials, see for example
...

**0**

votes

**0**answers

42 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 ...

**0**

votes

**0**answers

48 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

**0**answers

52 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 $I$ ...

**0**

votes

**0**answers

161 views

### Inertia/Gravity in Distance Geometry

The Cayley-Menger Determinant, D(N), slickly calculates the N-dimensional simplex
volume of any N+1 points. One constraint in our 3D world is that D(4)=0.
Give each point a mass (Mi) and dynamic ...