**9**

votes

**1**answer

396 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 ...

**0**

votes

**1**answer

171 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 ...

**-2**

votes

**1**answer

420 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:
...

**10**

votes

**2**answers

532 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

**3**answers

368 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

**1**answer

297 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

**1**answer

138 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 ...

**4**

votes

**1**answer

220 views

### Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)

Given correlation matrix $B$ (positive semi-definite with ones in the diagonal).
1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$.
2)Find the correlation matrix $A$ which ...

**2**

votes

**3**answers

334 views

### On matrices in linear forms with vanishing determinant

This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...

**7**

votes

**2**answers

450 views

### Is this a metric on the Grassmannian Manifold?

Let $m>n$ and consider the Set
$$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$
Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by
...

**4**

votes

**1**answer

277 views

### Determinant inequality of square-product sum of diagonal matrix and upper-triangular matrix

Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows.
For an arbitrary $n\times n$ diagonal matrix $\mathbf{D}$ and an arbitrary upper-triangular ...

**13**

votes

**2**answers

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 ...

**5**

votes

**2**answers

549 views

### Determinant of non-symmetric sum of matrices

Given three real, symmetric matrices $A\succ0$ and $B$, $C⪰ 0$.
How can it be shown that:
$$\det(A^2+AB+AC) \leq \det(A^2 +BA +AC+BC) ? \qquad (\star)$$
Where $A^2$ is symmetric and positive ...

**4**

votes

**3**answers

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:
...

**0**

votes

**1**answer

138 views

### Find a generalized hypergeometric-based function yielding certain ratios of fifth-degree polynomials

Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials)
\begin{equation}
...

**7**

votes

**1**answer

480 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 ...

**9**

votes

**1**answer

499 views

### M-matrix plus S-matrix is P-matrix?

I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two matrices and I would ...

**1**

vote

**1**answer

3k views

### Derivative of log determinant and inverse.

Hi all
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 evaluated
...

**0**

votes

**1**answer

209 views

### Gel'fand Yaglom functional determinant of non-diagonal operator?

Introduction:
As a quick reminder, the Gel'fand Yaglom theorem uses the generalized zeta-function approach to compute functional determinants of differential operators. Given a differential operator ...

**1**

vote

**0**answers

142 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 ...

**4**

votes

**0**answers

558 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;
$a_{ij}=0$ ...

**6**

votes

**1**answer

471 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 ...

**5**

votes

**4**answers

2k views

### Proving a determinant = 0

The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...

**15**

votes

**3**answers

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 ...

**2**

votes

**1**answer

193 views

### Endomomorphisms of Chain Complexes of vector spaces and determinants

Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$.
And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and $g_{\ast} : ...

**2**

votes

**0**answers

281 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 ...

**0**

votes

**1**answer

284 views

### Bounding a determinant ratio

Let $A=[A_{0}\ E;E^{T} \ B]$ be a real positive definite matrix and let $B$ be a principal submatrix. I am interested in tightly bounding $\frac{|B|}{|A|}$ from below in some "explicit" way that will ...

**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 ...

**6**

votes

**1**answer

623 views

### Determinantal formula for the nullspace of a singular matrix

In June 2012, Bill Press and Freeman Dyson published a remarkable paper on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that I feel I should ...

**2**

votes

**0**answers

136 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, ...

**15**

votes

**2**answers

652 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

**1**answer

444 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?

**2**

votes

**2**answers

457 views

### Generalizations of Oppenheim's inequality

The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$.
There has been a lot of beautiful work done ...

**10**

votes

**1**answer

4k views

### Sarrus rules for 4 times 4

This question is most probably not research level, but I thought that the MO folks might like it... Feel free to close.
Here is the motivation: If you have ever teached a maths course for engineers ...

**7**

votes

**2**answers

354 views

### Computing determinants of matrices of linear forms

Suppose we have three $n \times n$ matrices $A$, $B$, $C$ with floating point entries. We would like to compute the polynomial $\det (xA+yB+zC)$. At least in Mathematica, and I think in all computer ...

**4**

votes

**2**answers

715 views

### Is there a simple relation between the entropy of a matrix and its characteristic polynomial?

Assume $M$ is an invertible positive matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues $\{ \lambda_n \}$ is
$H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$.
In ...

**1**

vote

**0**answers

164 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
...

**2**

votes

**2**answers

281 views

### determinantal identity sought

Suppose $A$ is a $n \times m$ matrix and $B$ is a $m \times n$ matrix. Then it is known that $det(I_{n}+AB)=det(I_{m}+BA)$.
Is there an analogous identity of the form $det(P_{1}+AB)=det(P_{2}+BA)$, ...

**8**

votes

**2**answers

1k views

### Integral representation of a determinant

In a paper by Mathai, he uses the following integral representation of a determinant,
(or, really, what I give is a simple special case of what he gives), without any explication.
All matrices are ...

**4**

votes

**1**answer

297 views

### To compute minors of Jacobian of symmetric polynomials

For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$
one has Jacobian, expressed by the $(n \times n)$-determinants:
$$
...

**4**

votes

**1**answer

893 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
...

**1**

vote

**2**answers

195 views

### Does integral of A^T(t)*A(t) converge if det(A(t)) does not converge to 0?

Suppose you have an nxn parametric square matrix A(t). I am wondering if I can prove this:
$(lim\_{t\rightarrow\infty}det(A(t)) \ne 0) \Rightarrow (\int^{\infty}A^T(t)A(t)dt$ Does not Converge $)$

**0**

votes

**0**answers

163 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 ...

**1**

vote

**1**answer

286 views

### Identifying factors of higher order in a determinant

Consider a $n\times n$ matrix $A$ whose elements are some polynomials in the indeterminates $x_1, x_2,\ldots,x_m$. To calculate the determinant of such a matrix, one of the usual ways is to treat the ...

**13**

votes

**1**answer

939 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

**1**answer

421 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 ...

**9**

votes

**2**answers

878 views

### Extremal properties of the determinant for matrices with entries in a fixed subset of $[-1,1]^{n^2}$?

Given a multiset $S\subset [-1,1]^{n^2}$, we set
$$m(S)=\min\vert \det(M)\vert$$
where the minimum is over all matrices with entries forming the multiset $S$
and
$$a(n)=\max m(S)$$
where the maximum ...

**9**

votes

**2**answers

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 ...

**13**

votes

**4**answers

2k views

### determinant of the table of characters

I am certain that the answer to this question exists somewhere. It might be a classical exercise.
Let $G$ be a finite group. Its table of characters is a square matrix, whose rows are indexed by the ...

**20**

votes

**3**answers

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 ...