**57**

votes

**19**answers

19k 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. ...

**44**

votes

**7**answers

3k views

### How to prove this determinant is positive?

Given the matrices $ A_i=
\biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)
$, where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove that $\det(I + ...

**22**

votes

**3**answers

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 ...

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

**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 ...

**19**

votes

**6**answers

3k views

### Expressing adj(A) as a polynomial in A?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n ...

**19**

votes

**1**answer

492 views

### How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity
To ...

**18**

votes

**2**answers

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

**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 ...

**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
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...

**13**

votes

**4**answers

388 views

### Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant
$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i)
$$
over all $a_0,\dots,a_{n-1}$ such ...

**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 ...

**13**

votes

**2**answers

783 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

**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 ...

**13**

votes

**1**answer

940 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 ...

**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
$?

**10**

votes

**2**answers

426 views

### Are bounds known for the maximum determinant of a (0,1)-matrix of specified size and with a specifed number of 1s?

The problems of determining the maximum determinant of an $n \times n$ $(0,1)$-matrix and the spectral problem of determining exactly which other determinants can possibly occur are both reasonably ...

**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 ...

**10**

votes

**3**answers

687 views

### A class of matrix determinants between Wronskians and Vandermondes

Update: see below
Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions ...

**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 ...

**10**

votes

**1**answer

236 views

### Factor a sum of products of cofactors

Let $M$ be any $n\times n$ matrix.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$.
We can write ...

**9**

votes

**2**answers

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 ...

**9**

votes

**2**answers

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

**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 ...

**9**

votes

**1**answer

393 views

### Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...

**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 ...

**9**

votes

**2**answers

296 views

### Determinant of a checkerboard Hankel matrix with Catalan numbers

My goal is to compute
\begin{equation}
I = \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
...

**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

**1**answer

397 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 ...

**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 ...

**8**

votes

**0**answers

177 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

**2**answers

682 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

**3**answers

400 views

### An infinite product associated with random matrices

Motivation
Let ${\mathbb F}_q$ be the field with $q$ (a power of some prime number) elements. Then the order of $GL_n({\mathbb F}_q)$ is
$$(q^n-1)(q^n-q)\cdots(q^n-q^{n-1}).$$
The fact that this ...

**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
...

**7**

votes

**1**answer

371 views

### Block Matrix determinant

Consider the $k \times k$ block matrix:
$ C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \ddots & ...

**7**

votes

**1**answer

771 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 ...

**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 ...

**7**

votes

**1**answer

118 views

### Determinant of some covariance matrix (Gaussian kernel process)

Let $x_1,\dots,x_p$ be $p$ points in $\mathbb{R}^n$ ($n\geq 2$) with $x_1=0$. Consider the symmetric matrix $M(x)=(m_{ij}(x))_{1\leq i,j\leq p}$ where $m_{ij}(x) = \exp(-\frac{1}{2}\Vert x_i - ...

**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?

**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.
...

**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 ...

**6**

votes

**3**answers

2k 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

**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 ...

**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 ...

**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 ...

**6**

votes

**1**answer

94 views

### Sum identities with immanants

For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show
$$
\sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} ...

**6**

votes

**0**answers

203 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

**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 ...

**5**

votes

**2**answers

952 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

**1**answer

1k 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 ...