# Questions tagged [determinants]

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

447
questions

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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,...

15
votes

4
answers

879
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### 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:
...

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0
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71
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### 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....

6
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0
answers

134
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### 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 ...

5
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### 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 $d_i(A)$ ...

0
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0
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50
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### 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)...

5
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1
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140
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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 ...

2
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0
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127
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### 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 ...

1
vote

0
answers

100
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### 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 ...

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0
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107
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### 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 ...

3
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144
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### 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 ...

5
votes

1
answer

235
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### 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 ...

2
votes

2
answers

231
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### 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 ...

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### 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 ...

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101
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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 ...

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### Volume interpretation of number of spanning trees in planar graphs

The number of spanning trees of a planar graph is $DET$ complete where $DET$ class is given in https://complexityzoo.net/Complexity_Zoo:D#det by corollary $21$ in https://mbraverm.princeton.edu/files/...

2
votes

1
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### 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}
...

2
votes

0
answers

101
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### 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 ...

3
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154
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### 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 $ \...

1
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1
answer

203
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### 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=...

6
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0
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### 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 ...

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0
answers

170
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### 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 ...

4
votes

2
answers

148
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### 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 ...

2
votes

1
answer

126
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### 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 ...

1
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0
answers

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### 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 ...

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0
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23
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### Relation between optimizing determinant and optimizing the smallest eigen value

I have a question. Is there any relation between optimizing determinant and optimizing the smallest eigen value (lambda_min) of an n*n matrix?

1
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0
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### 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 ...

1
vote

1
answer

128
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### 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 ...

5
votes

1
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397
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### 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, ...

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0
answers

103
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### 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
$$\det\left[\tan\pi\frac{aj+bk}n\right]_{1\le j,k\le n-1}=\left(\frac{-ab}n\right)n^{n-2}$$
...

4
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0
answers

139
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### 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 ...

3
votes

1
answer

245
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### Generating functions for Hankel determinants of Catalan numbers

The Hankel determinants of the Catalan numbers are well known and can be written as
$d(k,n)= \det \left( C_{k + i + j} \right)_{i,j = 0}^{n - 1}=\prod_{i=1}^{k-1}\frac{\binom{2n+2j}{j}}{\binom{2j}{j}}$...

3
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2
answers

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### 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|...

2
votes

1
answer

320
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### 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, ...

5
votes

1
answer

370
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### 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 \...

1
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0
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29
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### 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 ...

19
votes

2
answers

903
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### 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}\...

15
votes

2
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### 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 $2 \times 2$ ...

5
votes

2
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248
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### 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+...

0
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1
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107
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### $\mathbb R$ 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+...

9
votes

1
answer

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### 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 ...

4
votes

3
answers

282
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### 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 ...

0
votes

1
answer

122
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### 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{...

0
votes

2
answers

99
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### 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 \...

4
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0
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72
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### 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]...

4
votes

2
answers

147
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### 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 ...

1
vote

1
answer

119
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### 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 $i\in \{1,5 \}$, ...

2
votes

1
answer

74
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### 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 $X$ ...

6
votes

0
answers

150
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### 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 ...

3
votes

2
answers

181
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### A problem about determinant and matrix

Suppose $a_{0},a_{1},a_{2}\in\mathbb{Q}$, such that the following determinant is zero, i.e.
$
\left |\begin{array}{cccc}\\
a_{0} &a_{1} & a_{2} \\
\\
a_{2} &a_{0}+a_{1} & a_{1}+a_{...