All Questions
Tagged with determinants reference-request
45 questions
23
votes
2
answers
3k
views
Formula expressing symmetric polynomials of eigenvalues as sum of determinants
The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
- 493
14
votes
1
answer
1k
views
Expansion of $\det(A+B)$
If $A,B\in{\bf M}_n(k)$, then the following formula holds true:
$$\det(A+B)=\sum_{r=0}^n\sum_{|I|=|J|=r}\epsilon(I,I^c)\epsilon(J,J^c)A\binom IJ B\binom{I^c}{J^c}.$$
In this formula, $I$ and $J$ are ...
- 52.3k
11
votes
3
answers
918
views
yet another determinant and inverse of a matrix
This problem is some variation of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c &...
- 43.2k
10
votes
1
answer
520
views
Homogeneous polynomials, mixed determinants, positive definiteness
Are there $n\times n$ real matrices $A_{1}, \ldots, A_{n}$ such that the $n$-homogeneous polynomial
$$
f(x_{1}, \ldots, x_{n}) = \det(x_{1} A_{1}+\cdots +x_{n} A_{n})
$$
never vanishes on $\...
- 3,946
9
votes
2
answers
3k
views
Integral representation of a determinant
In a paper by Mathai, he uses the following integral representation of a determinant,
(or, really, what I give is a simple special case of what he gives), without any explication.
All matrices are ...
- 2,600
9
votes
2
answers
495
views
Is there a determinantal point process proof of the Keating-Snaith formula for the cumulants of the log characteristic polynomial of a random matrix?
For $U$ a unitary $N \times N$ matrix, randomly distributed according to Haar measure, we have the complex-valued random variable $\log (\det (1-U))$. The real part and imaginary parts of $\log (\det (...
- 148k
7
votes
3
answers
1k
views
Determinant of correlation matrix of autoregressive model
I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form
$$R = \begin{pmatrix}
1 & r & \cdots & r^{d-...
- 719
7
votes
0
answers
444
views
Determinants associated to orthogonal polynomials
Let $${p_n}(x) = \sum\limits_{j = 0}^{n } {{{( - 1)}^{n - j}}p(n,j){x^j}} $$ be orthogonal polynomials satisfying $${p_n}(x) = (x - {s_{n - 1}}){p_{n - 1}}(x) - {t_{n - 2}}{p_{n - 2}}(x)$$ with ...
- 5,622
6
votes
2
answers
1k
views
Determinant of the oriented adjacency matrix of a tree
Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries
$$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\
-1 &...
- 27.8k
6
votes
0
answers
392
views
Divisibility properties of minors of matrices
Let $A$ be an $m\times n$ matrix with integer entries. Let $d_i(A)$ be the greatest common divisor of all $i\times i$ minors of $A$, and define $d_0(A)=1$. Whenever $i\leq j$, one has that $d_i(A)$ ...
- 191
6
votes
0
answers
252
views
Does anyone know this determinant?
The following determinant arises in a combinatorial enumeration problem. I wonder if anyone has seen it before in any context or knows how to evaluate it. I tried computing it for small $n$ but didn't ...
- 37.7k
5
votes
1
answer
251
views
Hankel determinants for q-Catalan numbers where q is a root of unity?
Let ${C_n}(q)$ be the weight of the Dyck paths of semilength $n$ where the upsteps have weight $1$ and the downsteps which end on height $i$ have weight $q^i$.
They satisfy ${C_n}(q) = \sum\limits_{j ...
- 5,622
5
votes
1
answer
126
views
Identity relating iterated determinant line bundles
Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
- 2,356
5
votes
1
answer
312
views
Expected inverse determinant with independent rows
Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors.
More ...
- 1,533
5
votes
0
answers
190
views
Yet, another generalization of Catalan determinants
The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
- 43.2k
4
votes
3
answers
369
views
Determinant in terms of certain $2\times 2$ minors
Let $A$ be an $n\times n$ matrix with entries $a_{i,j}$. Define an $(n-1)\times(n-1)$ matrix $B$ with entries $b_{i,j}=a_{1,1}a_{i+1,j+1}-a_{1,j+1}a_{i+1,1}$. Then $\det(B)=a_{1,1}^{n-2}\det(A)$.
I ...
- 191
4
votes
1
answer
781
views
Determinant of a random row stochastic matrix
Does anyone know anything about the determinant of a random $n\times n$ row stochastic matrix? What I have in mind is that the rows are independently selected from the uniform distribution on the unit ...
- 23.2k
4
votes
1
answer
964
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
$$
\lim_N\det(...
- 2,135
4
votes
0
answers
181
views
Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
4
votes
0
answers
163
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.2k
4
votes
0
answers
96
views
Bessel in matrix?
Let $M_n$ be the matrix
$$M_n=\begin{pmatrix}
1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\
1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
- 43.2k
3
votes
0
answers
185
views
"Circulant-Vandermonde" matrix: in search of a formula
An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...
- 43.2k
3
votes
0
answers
207
views
On a variation of the Vandermonde matrix
The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant
$$\prod_{i<j}^{1,n}(x_j-x_i)$$
have found many utilities in Combinatorics and Physics, among other ...
- 43.2k
3
votes
0
answers
133
views
Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)
For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
- 131
3
votes
0
answers
105
views
Can one do better than using general purpose determinant algorithms when using the Fisher-Kasteleyn-Temperley method for perfect matchings?
Questions.
(numerical.generalPfaffian) Is it proved anywhere that in general it is not easier0 to calculate the determinant (over $\mathbb{Q}$) of the skew-symmetric signed adjacency matrix defined ...
- 6,051
3
votes
0
answers
281
views
How do I obtain Vandermonde identity from Weyl's denominator formula?
Let $\Phi$ be a root system with Weyl group $\operatorname{Weyl}(\Phi)$, let $\Phi^+$ be a set of positive roots for $\Phi$ and $\rho$ be the half sum of the elements of $\Phi^+$. Then the Weyl's ...
- 1,479
3
votes
0
answers
130
views
Where does this identity involving sums of Hankel-like determinants over partitions come from?
For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix $$M_f(\...
- 13.4k
2
votes
2
answers
3k
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,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 7,442
2
votes
1
answer
305
views
Some determinants which are closely related to recurrences
Let the sequence $(a(n,k))_{ n \in \mathbb{Z}}$ satisfy $$\sum_{j=0}^k c(k,j)a(n-j,k)=0$$ with $c(k,j)=c(k,k-j)$ and $c(k,0)=1$ and with initial values $a(-n,k)=0$ for $1\leq n\leq{k-1}$ and $a(0,k)=...
- 5,622
2
votes
1
answer
417
views
Roots of determinant of matrix with polynomial entries
Let $p_1, p_2,\dots, p_n$ and $q_1,q_2,\dots,q_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying
$$a_{ij} = \begin{cases} p_i(x) & \text{ if } i = j, \\ q_i(x)...
- 1,269
2
votes
1
answer
214
views
A Vandermonde like determinant with exponentials
Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\...
- 275
2
votes
1
answer
325
views
Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
- 13.4k
2
votes
0
answers
130
views
Pfaffian generalization
The identity
$$\left|
\begin{array}{cccc}
x & y_1 & y_2 & y_3 \\
z_1 & 0 & a & b \\
z_2 & -a & 0 & c \\
z_3 & -b & -c & 0 \\
\end{array}
\right|=\...
- 12.3k
2
votes
0
answers
233
views
Roots of determinant of matrix with polynomial entries — a generalization
For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...
- 1,269
2
votes
0
answers
276
views
An (open?) problem about a sequence of nested principal sub-matrices and their determinants
Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
- 59
1
vote
1
answer
76
views
Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
- 20.2k
1
vote
1
answer
254
views
When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?
Let $A$ be an $n \times n$ real symmetric matrix.
Let
$$
M = \begin{pmatrix} A & X \\ X^T & A \end{pmatrix}
$$
where $X$ is a real invertible $n \times n$ matrix. I am interested in finding ...
- 3,625
1
vote
0
answers
210
views
Algebraic relation given by a 3x3 determinant
I just encountered a very curious relation in an algebra. A bit simplified, I am working in a (particular) non-commutative algebra, with some relations.
One particular relation is the following:
For (...
- 15.8k
1
vote
0
answers
159
views
Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?
Disclaimer: This might be an SE question, but I'm not quite sure...
Thanks in advance!
Setup
So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
- 6,853
1
vote
0
answers
148
views
Traces in associative algebras
Are there some books or papers about the general definition of traces:
If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...
- 283
1
vote
0
answers
114
views
Has this logarithmic volume functional been studied?
$\newcommand{\M}{\mathcal{M}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\VolM}{\text{Vol}_{\M}}
\newcommand{\VolN}{\text{Vol}_{\N}}$
This question is mainly a reference request. (It is a cross-post ...
- 6,741
0
votes
1
answer
129
views
A variant of numeric Vandermonde which failed symbolically?
Given some variables $x_1, x_2, \dots, x_n$, the Vandermonde determinant is given by
$$V_n(x_1,\dots,x_n):=\det(x_j^{i-1})_{i,j=1}^n=\prod_{i<j}(x_j-x_i).$$
One can take as special cases: $x_j=j$ ...
- 43.2k
0
votes
1
answer
525
views
What is the mathematician's definition of the determinant? [closed]
I am trying really hard to find a good definition of the determinant.
I have looked virtually every single resource online and everybody gives a different answer:
sum of cofactors or minors https://...
- 179
0
votes
0
answers
282
views
A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular
A paper I'm reading in representation theory states the following result:
Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
- 6,180
-1
votes
2
answers
1k
views
A sum-of-determinants identity [closed]
I posed a terser version of this question on math.stackexchange.com and after 24 hours I get only a comment on a detail of notation and neither votes nor answers.
Suppose $X_1,\ldots,X_{n+1}\in\...
