Questions tagged [determinants]

Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.

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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-...
T. Amdeberhan's user avatar
7 votes
2 answers
772 views

Determinant of matrix with Stirling numbers as elements

After noticing that the determinant of an $n \times n$ matrix $A_n$ with elements $a_{i,j}=i^j$, $1 \le i \le n$, $1 \le j \le n$, is the superfactorial (product of the first $n$ factorials), I wanted ...
Fabius Wiesner's user avatar
2 votes
0 answers
182 views

$\det(I+A) \leq 1$ with equality if and only if $A$ is nilpotent

Suppose we are given the following: An $n \times n$ matrix $A$ (with rational entries). $\operatorname{trace}(A)=0$ $B=A+I$ where $I$ is the $n \times n$ identity matrix. Question. What conditions ...
Anonymous's user avatar
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Determinant of barycenter of a hyperbolic-matrix

Let $A \in \operatorname{GL}(d, \mathbb{R})$ be a hyperbolic matrix. I want to show that $$\det((1-\alpha)A+\alpha\operatorname{Id})\geq 1,$$ where $0<\alpha<1$. Attempt: In $\operatorname{SL}(2,...
Adam's user avatar
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17 votes
4 answers
2k views

Possible values of the determinant for matrices with elements $\{1, 0, -1\}$

For matrices with elements $\{-1, 1\}$ it is known from here that the possible absolute values of determinants of $n \times n$, $n \leq 6$ matrices with entries $\{-1, 1\}$ are as follows: ...
Konrad Burnik's user avatar
1 vote
0 answers
100 views

Determinant Inequality with unitary matrix

I come up with the following conjecture while doing my research, which is a determinant inequality. I have tried to run the MatLab simulation to verify its sanity. It seems that the inequality is true....
Quicky2357's user avatar
6 votes
0 answers
213 views

Lindström-Gessel-Viennot from properties of the $Alt^k$ functor?

Let $A$ be the directed adjacency matrix of an acyclic directed graph, with variables as its nonzero entries (one for each edge). The $(a,b)$ entry of the matrix $(I-A)^{-1}$ is the sum over all paths ...
Allen Knutson's user avatar
5 votes
0 answers
339 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 $d_i(A)$ ...
Joel Louwsma's user avatar
0 votes
0 answers
87 views

Number of roots of a Vandermonde like complex determinant

I am originally interested in the determinant $$ \left|\begin{array}{cccc}\exp(i\lambda_1\cdot x_1) & \exp(i\lambda_2\cdot x_1) & ... & \exp(i\lambda_n\cdot x_1) \\\exp(i\lambda_1\cdot x_2)...
kaleidoscop's user avatar
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5 votes
1 answer
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Log determinant of quadratic form

I am reading a paper Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction where the author uses the following result from matrix analysis but does not explain why it is true nor provide ...
lmn2609's user avatar
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3 votes
0 answers
188 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 ...
T. Amdeberhan's user avatar
1 vote
0 answers
118 views

Determinants associated with Stern's diatomic sequence

Consider the so-called Stern's triangle (refer to these slides by R. Stanley), we denote here by $a_n(k)$. In an article Some linear recurrences motivated by Stern’s diatomic array, Stanley provided ...
T. Amdeberhan's user avatar
1 vote
0 answers
132 views

Hankel determinants of the sequence $(\binom{n}{m})_{n\ge0}$ and related sequences

I posted (https://math.stackexchange.com/questions/4363151/generating-functions-of-hankel-determinants-related-to-hoggatt-triangles) this question on Mathematics StackExchange but have not received a ...
Johann Cigler's user avatar
3 votes
0 answers
240 views

Learning about determinantal varieties

In my research I recently stumbled upon a problem which involves trying to identify whether a given projective variety is determinantal or, even stronger, determinantal of a particular form. For ...
Sergey Guminov's user avatar
5 votes
1 answer
248 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 ...
Johann Cigler's user avatar
2 votes
2 answers
282 views

Determinants of minors occurring 'within' determinant of full matrix

$A= (a_{ij})$ is an $n\times n$ symmetric positive matrix. It induces a quadratic form $f(x):= x^tAx$ on $\mathbb{R}^n$. $D_m$ denotes the determinant of the top left $m\times m$ submatrix of $A$ (or ...
Chertopkhanov on Malek Adel's user avatar
1 vote
0 answers
65 views

On perfect matchings on planar graphs - is there a linear time deterministic algorithm?

The slides here provide a way to get a pfaffian orientation from Minimum Spanning Tree. MST can be found in linear time if graph is planar and weights are $1$ and the slides give a linear time ...
Turbo's user avatar
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Determinant of chain complexes

Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of ...
Ronald J. Zallman's user avatar
2 votes
1 answer
159 views

Monotonicity of the determinant of symmetric Toeplitz Matrices

For simplicity, i focus on a particular Toeplitz symmetric matrix, so let $A_{ij} = a^{|i-j|}$ for $i,j=1,\dots,n$ and $0<a<1$ be a Kac-Murdock-Szegő (KMS) matrix, e.g., for n=4 \begin{equation} ...
Andrea Tani's user avatar
2 votes
0 answers
110 views

Can the absolute value of fixed sized minors be arbitrarily ordered?

In an $m \times n$ matrix $X$, there are exactly $\binom{m}{r}\binom{n}{r}$ minors of size $r \times r$. Is it always possible to construct a real matrix $X$ such that the absolute value of the ...
Powerspawn's user avatar
3 votes
0 answers
208 views

Combinatorial interpretation of a determinant

This is a continuation of Worpitzky-like identities?. Let $ r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$ As Sam Hopkins has remarked $r_k(x)$ is the number of plane partitions in a $ \...
Johann Cigler's user avatar
1 vote
1 answer
247 views

Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible

$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix: $$ M:= \begin{bmatrix} \cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\ \cPr{X=...
G. Ander's user avatar
6 votes
0 answers
74 views

Rank of matrix coming from cobordism computations

In a computation of Pontryagin-numbers of certain manifolds (see the appendix of https://arxiv.org/pdf/2109.10306.pdf for more context) we came across the following elementary problem: Consider the ...
Georg Frenck's user avatar
1 vote
0 answers
207 views

How to calculate Toeplitz-type determinant expansion?

We want to calculate next sum in different point in limit of large $N, N_f$. Initial expression is expansion around $h=0$ by definition. And the answer is known . This is ($N_f/N= \kappa$) $$ \lim_{N ...
Sergii Voloshyn's user avatar
4 votes
2 answers
184 views

Computation of the pfaffian of a particular matrix

This question was originally asked in (https://math.stackexchange.com/questions/4265063/computation-of-the-pfaffian-of-a-particular-matrix). I did not find any satisfactory answer there. So I am ...
SiOn's user avatar
  • 493
2 votes
1 answer
136 views

Existence of matrices with some invertibility properties

Prove that there exists five matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$. I am interested ...
user avatar
1 vote
0 answers
87 views

Evaluate $\det[[\lfloor\frac{aj-(a+1)k}n\rfloor]_q]_{1\le j,k\le n}$ and $\det[[\lceil\frac{(a+1)j-ak}n\rceil]_q]_{1\le j,k\le n}$

The $q$-analogue of an integer $m$ is defined by $[m]_q=(1-q^m)/(1-q)$. Note that $\lim_{q\to1}[m]_q=m$. I have formulated the following conjecture on determinants involving the floor function and the ...
Zhi-Wei Sun's user avatar
  • 14.4k
1 vote
0 answers
80 views

How to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$? [closed]

Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$. I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$. My belief is that this is true is motivated by empirical ...
Msc Splinter's user avatar
1 vote
1 answer
166 views

Properties of the generic matrix - struggles with constructive proofs

Write $A=(x_{ij})$ for the generic matrix (comprised of indeterminates) defined over $\mathbb Z[x_{11},\dots,x_{nn}]$. In their constructive commutative algebra book, Lombardi and Quitte write that ...
Arrow's user avatar
  • 10.3k
6 votes
1 answer
506 views

A novel identity connecting permanents to Bernoulli numbers

For a matrix $[a_{j,k}]_{1\le j,k\le n}$ over a field, its permanent is defined by $$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}:=\sum_{\pi\in S_n}\prod_{j=1}^n a_{j,\pi(j)}.$$ In a recent preprint of mine, ...
Zhi-Wei Sun's user avatar
  • 14.4k
1 vote
1 answer
215 views

A determinant involving the cotangent function

Let $n>1$ be odd. In my 2019 preprint On some determinants involving the tangent function, I proved that $$\det\left[\tan\pi\frac{aj+bk}n\right]_{1\le j,k\le n-1}=\left(\frac{-ab}n\right)n^{n-2}$$ ...
Zhi-Wei Sun's user avatar
  • 14.4k
10 votes
1 answer
531 views

Identities involving derangements and roots of unity

For a positive integer $n$, a derangement of $\{1,\ldots,n\}$ is a permutation $\tau$ of $\{1,\ldots,n\}$ with $\tau(j)\not=j$ for all $j=1,\ldots,n$. For convenience, we let $D(n)$ denote the set of ...
Zhi-Wei Sun's user avatar
  • 14.4k
4 votes
1 answer
394 views

Generating functions for Hankel determinants of Catalan numbers

The Hankel determinants of the Catalan numbers are well known and can be written as $d(k,n)= \det \left( C_{k + i + j} \right)_{i,j = 0}^{n - 1}=\prod_{i=1}^{k-1}\frac{\binom{2n+2j}{j}}{\binom{2j}{j}}$...
Johann Cigler's user avatar
3 votes
2 answers
246 views

Values of the determinants $\det[|j-k|]_{1\le j,k\le n}$ and $\det[|j^2-k^2|]_{1\le j,k\le n}$

Based on my computation, I have formulated the following conjecture. Conjecture. For any positive integer $n$, we have $$\det[|j-k|]_{1\le j,k\le n}=(-1)^{n-1}(n-1)2^{n-2}\tag{1}$$ and $$\det[|j^2-k^2|...
Zhi-Wei Sun's user avatar
  • 14.4k
2 votes
1 answer
361 views

Determinants of striped Hankel matrices

This question is related to the matrices described in Deyi Chen's recent MO post (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, ...
T. Amdeberhan's user avatar
5 votes
1 answer
394 views

An interesting Hankel determinant

Let $h(n,t) = \sum\limits_{j = 0}^n {\binom {\lfloor {\frac{n}{2}} \rfloor }{j}\binom {\lfloor {\frac{n+1}{2}}\rfloor }{j}t^j \\ }.$ I am interested in the Hankel determinants $${D_k}(n,t) = \det \...
Johann Cigler's user avatar
2 votes
0 answers
59 views

What is the distribution of determinant of multi multiplication of some Gaussian matrices?

I have a square matric $H = (ABC)(ABC)^H$ where $A$ and $C$ are complex Gaussian matrices with some correlation matrices and $B$ is a diagonal matrix with entries $e^{j \theta}$ on the diagonal such ...
Mahdi Eskandari's user avatar
20 votes
2 answers
1k views

Connection between determinant and quotient rule

For the function $\dfrac{f(x)}{g(x)}$, we have, $\left(\dfrac{f}{g}\right)' = \dfrac{gf'-fg'} {g^2}$. We can write the numerator as $W(g,f) = \left|\begin{matrix} g & f \\ g' & f'\end{matrix}\...
Etemon's user avatar
  • 385
16 votes
2 answers
2k views

Proof that block matrix has determinant $1$

The following real $2 \times 2$ matrix has determinant $1$: $$\begin{pmatrix} \sqrt{1+a^2} & a \\ a & \sqrt{1+a^2} \end{pmatrix}$$ The natural generalisation of this to a real $2 \times 2$ ...
eaglebrain's user avatar
5 votes
2 answers
275 views

General formulas for derivative of $f_n(x)=\dfrac{ax^n+bx^{n-1}+cx^{n-2}+\cdots}{a'x^n+b'x^{n-1}+c'x^{n-2}+\cdots},\quad a'\neq0$

For the function $f_1(x)=\dfrac{ax+b}{a'x+b'},\quad a'\neq0$ , we have $$f_1'(x)=\dfrac{\begin{vmatrix}{a} && {b} \\ {a'} && {b'}\end{vmatrix}}{(a'x+b')^2}$$ For $f_2(x)=\dfrac{ax^2+bx+...
Etemon's user avatar
  • 385
0 votes
1 answer
229 views

$\mathbb R$ and $\mathbb F_2$ rank in boolean matrix product

By rank I imply rank over reals ($\mathbb R$). I consider two $n\times n$ matrices $A,B$ having entries in $0/1$. The product below follows usual matrix product rules except $xy$ is $AND(x,y)$ and $x+...
User2021's user avatar
10 votes
1 answer
614 views

Is it true that $\det\big[\sin 2\pi\frac{(j-k)^2}p\big]_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}}{2^{p-1}}$ for each prime $p\equiv3\pmod4$?

Question. Does the equality $$\det\left[\sin 2\pi\frac{(j-k)^2}p\right]_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}}{2^{p-1}} $$ hold for every prime $p\equiv3\pmod4$? I have checked the equality numerically ...
Zhi-Wei Sun's user avatar
  • 14.4k
4 votes
3 answers
335 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 ...
Joel Louwsma's user avatar
1 vote
1 answer
210 views

Solving recurrence relation for symmetric Toeplitz matrices determinant

$$\text{Let } T_n=\begin{Bmatrix} a & b & \boldsymbol{0} \\ b & a & \ddots \\ \boldsymbol{0} & \ddots & \ddots \end{Bmatrix}\text{...
Aileann D. PRET's user avatar
0 votes
2 answers
102 views

Computation to differentiate a determinant [closed]

Consider a positive Hermitian $N \times N$ matrix $A$ with complex valued coefficients. We list its eigenvalues in increasing order and with their multiplicities, $\mu_{1} \leq \mu_{2} \leq \cdots \...
NSR's user avatar
  • 97
4 votes
0 answers
78 views

Minimal set generators ideal submaximal minors

Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as: $$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
Libli's user avatar
  • 7,210
4 votes
2 answers
196 views

Hankel determinant of incomplete gamma functions

I have some expressions that involve Hankel determinants of incomplete gamma functions. They are of the ($r \times r$ form) I'd like to evaluate these determinants. Elementary operations help, but ...
searp's user avatar
  • 41
1 vote
1 answer
174 views

Existence of matrices in the field $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$ and let $e_1$, $e_2$, $\ldots$, $e_{10}$ denote its rows. For $i\in \{1,5 \}$, ...
user avatar
2 votes
1 answer
81 views

Existence of a matrix in $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$. What are all the possible $n$ ($\geq 6$) for which there exists a matrix $X$ ...
user avatar
6 votes
0 answers
466 views

Expressing a polynomial as the determinant of a matrix of linear forms

I have heard that it's a well known result (in theoretical computer science?) that if we have a polynomial $p(t_1,\dots,t_n)$ over $\mathbb Q$, we can find matrices $M_0,\dots,M_n/\mathbb Q$ such that ...
Asvin's user avatar
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