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53
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. ...
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 ...
20
votes
4answers
954 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: $perm(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, \sigma(i)}$ ...
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 ...
17
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 ...
15
votes
3answers
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 ...
14
votes
2answers
590 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 $$ \det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB) $$ ...
13
votes
2answers
756 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 ...
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 ...
12
votes
2answers
1k 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 ...
10
votes
2answers
496 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 $?
10
votes
2answers
884 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 ...
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 ...
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 ...
9
votes
1answer
397 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 ...
9
votes
1answer
2k 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 ...
9
votes
2answers
813 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
1answer
252 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 ...
9
votes
0answers
515 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 ...
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 ...
8
votes
2answers
748 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 ...
7
votes
2answers
661 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?
7
votes
2answers
302 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 ...
7
votes
2answers
327 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 ...
7
votes
1answer
458 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 ...
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 ...
6
votes
1answer
723 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 ...
6
votes
1answer
234 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 ...
6
votes
1answer
426 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 ...
6
votes
1answer
407 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 ...
6
votes
0answers
126 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 ...
6
votes
0answers
456 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
446 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 ...
5
votes
2answers
893 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 ...
5
votes
1answer
808 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 ...
5
votes
4answers
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 ...
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 ...
5
votes
2answers
561 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 ...
5
votes
1answer
388 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 ...
5
votes
0answers
331 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?
4
votes
1answer
842 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 ...
4
votes
2answers
508 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 ...
4
votes
1answer
194 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 ...
4
votes
2answers
272 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 ...
4
votes
2answers
521 views

What's the correct notion of determinant of a bilinear pairing?

By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...
4
votes
1answer
195 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 ...
4
votes
1answer
253 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: $$ ...
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
3answers
2k 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: ...
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 $$ ...