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7
votes
2answers
333 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
2answers
549 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
0answers
156 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
2answers
273 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
2answers
812 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
1answer
262 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
1answer
850 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
2answers
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
0answers
156 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
1answer
281 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 ...
11
votes
0answers
559 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
1answer
393 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
2answers
833 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
2answers
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
4answers
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
3answers
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
1answer
488 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
0answers
459 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
2answers
569 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
1answer
269 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
1answer
473 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
1answer
843 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
2answers
767 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
3answers
651 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
1answer
594 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 ...
20
votes
4answers
994 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
2answers
916 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
2answers
668 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
1answer
663 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 ...
5
votes
1answer
847 views

Sum of squares of determinants of principal minors

I am interested in computing the sum of squares of determinants of principal minors. Let $A$ be an $n\times n$ positive semidefinite matrix and $A_S$ be a principal minor of $A$ indexed by the set $S ...
4
votes
2answers
274 views

analogue of GUE and Ginibre in higher dimensions

This is a completely unmotivated question, but what happens to the 1-point marginal distribution for the following $N$-point joint distribution: $$\displaystyle p(z_1,\ldots, z_N) = C_N ...
2
votes
1answer
240 views

Why doesn't the argument of circular law convergence of Ginibre spectrum give the same result for GUE?

It appears I am profoundly confused in the following nice argument of Ginibre and Mehta and beautifully presented in Djalil Chafai's blog ...
5
votes
2answers
906 views

universal property of the determinant bundle

Let $X$ be a nontrivial ringed space (i.e. all stalks are nonzero). To every locally free module $M$ on $X$ of constant rank $n$ we can associated it's determinant $\det(M)$, which is a line bundle ...
1
vote
1answer
106 views

Question on a relation between minors of a particular kind of matrix

Hi! Perhaps it is an easy question but i don't figure out how to prove it. Let $a_1,...,a_{2m+2}\in\mathbb{C}$ and for $1\leq i\leq 2m+2$ and $j\leq [\frac{2m+2-i}{2}]$ (with $[a]$ i mean the integer ...
2
votes
1answer
220 views

A determinant involving only cyclotomic factors

Let $\alpha:\mathbb Z\longrightarrow \mathbb Z$ be a quadratic polynomial taking only integral values on the integers and consider the sequence of square-matrices with coefficients $x^{\alpha(i+j)}$ ...
5
votes
1answer
253 views

Expected inverse determinant with independent rows

Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors. More ...
3
votes
1answer
816 views

Determinant and symmetric power

Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...
3
votes
3answers
2k views

What's the maximum determinant of the (0, 1) matrix from M(n, R)?

If there's no exact formula what's the nearest upper and lower bounds do you know?
3
votes
2answers
2k views

Matrix products under which the determinant behaves multiplicatively

The determinant behaves multiplicatively with respect to the usual matrix product $$ \det(AB) = \det(A)\det(B), $$ and also with respect to the Kronecker (or tensor) product of square matrices $$ ...
3
votes
0answers
768 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 ...
7
votes
1answer
742 views

Appropriate journal to publish a determinantal inequality

I have recently made the following observation: Let $v_i := (v_{i1}, v_{i2})$, $1 \leq i \leq k$, be non-zero positive elements of $\mathbb{Q}^2$ such that no two of them are proportional. Let ...
9
votes
2answers
2k views

Determinant of a 3x3 Magic Square

Hello guys. This is my first question with mathOverflow so I hope my etiquette is up to par here. My question is regarding a 3x3 magic square constructed using the la Loubere method (see la Loubere ...
54
votes
19answers
16k 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
0answers
249 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. ...
0
votes
1answer
504 views

Hankel determinants of symmetric functions

The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions. ...
19
votes
3answers
2k views

Splitting the determinant polynomial into linear factors - a Dedekind problem

Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial $\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq ...
8
votes
2answers
1k views

What are picard categories, where can I learn more about them, and why should I care to?

I have the category-theoretic background of the occasional stroll through Mac Lane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor ...
6
votes
3answers
1k views

Sarrus determinant rule: references, extensions

SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS An undergraduate came to me with an identity for 4x4 determinants that is actually correct: $\det(A)=h(A)+h(RA)+h(R^{2}A)$ where R cyclically ...
18
votes
2answers
1k views

Lifting matrices mod 2 to integers.

The following question was motivated by my research. Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs ...
2
votes
2answers
1k views

Statement of Lagrange's theorem on determinants(elementary question).

Apologies for this elementary question; but I was unable to find a reference otherwise. Let $A, B, C$ be square matrices of the same dimension. Then, $$\begin{vmatrix} A & C \\\ 0 & B ...