**3**

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

130 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

470 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

474 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

179 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

140 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

548 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

460 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

1k 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

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 ...

**0**

votes

**1**answer

257 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

264 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

587 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

133 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

634 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

423 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

423 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

3k 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

348 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

627 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

160 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

277 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

957 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

286 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

883 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

194 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

160 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

284 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

908 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

405 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

868 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

1k 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 ...

**1**

vote

**1**answer

494 views

### Counting matrices with different determinants

Let $A$ and $B$ be two matrices of order $n$ over a finite subset of integers $S$ such that $A$ and $B$ are positive-definite, nonsingular and symmetric.
I am interested in proprieties about $A$, $B$ ...

**7**

votes

**0**answers

471 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.
...

**5**

votes

**2**answers

587 views

### How can I write down a point in the Berezinian of a super vector space?

A vector space $V$ of dimension $n$ has an associated determinant line $Det(V)$.
An element of $Det(V)$ is represented as a (formal limear combination) of expresstions of the form
$v_1 \wedge ...

**1**

vote

**1**answer

273 views

### Encoding information about submatrix determinants

$M$ is an $n\times n$ matrix. Consider the submatrices $M(P;Q)$ formed from $P$ rows and $Q$ columns of $M$ where $P$ and $Q$ are disjoint indices.
Is there some way to encode the various ...

**3**

votes

**1**answer

499 views

### Convergence of Fredholm determinants

Let $(X_N)_N$ be a sequence of trace class operators acting on, say, $L^2(\mathbb{R})$. What are the minimal assumptions in order to have the convergence of their Fredholm determinant
$$
...

**2**

votes

**1**answer

920 views

### determinant of exterior power

Suppose A is a n times n matrix.
what is the determinant of the i-th exterior power of A, in terms of determinant of A ?
thanks..

**13**

votes

**2**answers

779 views

### How to invert the matrix [n choose 2j - i] ?

In a certain model of a stat-physics type, one encounters a matrix
$$
A_n:=\left[\binom{n}{2j-i}\right]_{i,j=1}^{n-1}.
$$
The determinant of this matrix (equal to $2^{\binom n2}$) counts the number ...

**1**

vote

**3**answers

661 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?

**1**

vote

**1**answer

643 views

### about Generalized Determinant?

Page 276 in the book Differential Topology and Quantum Field theory by C. Nash, describes a "generalization of determinant of linear map" as follows: for linear map
$O:{V} \to {W}$
its determinant ...

**21**

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:
$perm(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, \sigma(i)}$
...

**10**

votes

**2**answers

1k views

### Determinant associated with a group action

Let $G$ be a finite group and $S$ be a finite set, with $G$ acting on $S$. I consider indeterminates $x_g$ indexed by $g\in G$ and form the matrix of the group action $A\in M_{S\times S}$. Its entries ...

**7**

votes

**2**answers

679 views

### A Linear Algebra Question

Let $M$ be a symmetric square matrix with integer coefficients and $M_k$ the matrix obtained by deleting the k-th line and k-th column. If det(M)=0 does it follow that $\det(M_kM_j)$ is a square?

**2**

votes

**1**answer

689 views

### the inverse of determinant line bundle?

I am reading materials about the determinant defined by Knudsen-Mumford
http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=103495&vfpref=html&r=11&mx-pid=437541
which ...