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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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_{...
HilbertHunnterrrD's user avatar
8 votes
1 answer
279 views

On the determinant $\det[\gcd(i-j,n)]_{1\le i,j\le n}$

In Sept. 2013, I investigated the determinant $$D_n=\det[\gcd(i-j,n)]_{1\le i,j\le n}$$ and computed the values $D_1,\ldots,D_{100}$ (cf. http://oeis.org/A228884). To my surprise, they are all ...
Zhi-Wei Sun's user avatar
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3 votes
0 answers
407 views

What intuitive meaning "determinant" of a divergency (divergent integral or series) can have? [closed]

I am working on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\...
Anixx's user avatar
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2 votes
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For the following class of matrices, are the determinants invariant under permutations?

I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
Charles Cao's user avatar
2 votes
1 answer
247 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)=...
Johann Cigler's user avatar
1 vote
0 answers
108 views

Is there a bipartite graph whose determinant corresponds to number of perfect matchings?

Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$...
Turbo's user avatar
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5 votes
1 answer
122 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 ...
Owen Biesel's user avatar
  • 2,326
2 votes
0 answers
128 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|=\...
Alexey Ustinov's user avatar
1 vote
1 answer
392 views

A determinant identity

The following identity involving determinants essentially appears in E.L. Ince's book on Ordinary Differential Equations: Let $A$ be an $n \times n$ matrix, $n \geq 3$. Denote by $A_{j_1,\ldots,j_r}^{...
Vassilis Papanicolaou's user avatar
0 votes
0 answers
65 views

The minimum number of polynomial equations the components of linearly dependent vectors must satisfy

Context: Consider $m<n$ vectors $v_1,\dotsc,v_m\in\mathbb{C}^n$ with complex components. We can study if they are linearly dependent by constructing the following matrices. First the $n\times m$ ...
FeedbackLooper's user avatar
7 votes
1 answer
233 views

Reference for permanent integral identity

$\DeclareMathOperator\perm{perm}\DeclareMathOperator\diag{diag}$Using MacMahon's master theorem, the properties of complex gaussian integrals, and Cauchy's integral theorem one can show that the ...
motherboard's user avatar
2 votes
2 answers
271 views

Expectation of the determinant of the inverse of non-central Wishart matrix

Let $A$ be $(n,n)$ central Wishart matrix with $k$ degrees of freedom. my question is there is a way to estimate the expectation of: \begin{align} E[det(I+(I+A)^{-1})] \end{align}
hichem hb's user avatar
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20 votes
2 answers
2k 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 ...
Jules's user avatar
  • 463
5 votes
1 answer
312 views

Determinant of a certain Toeplitz matrix

Compute the following determinant \begin{vmatrix} x & 1 & 2 & 3 & \cdots & n-1 & n\\ 1 & x & 1 & 2 & \cdots & n-2 & n-1\\ 2 & 1 & x & 1 &...
LogicNotFound's user avatar
2 votes
1 answer
313 views

Two versions of Sylvester identity

MathWorld presents the following two versions of Sylvester's determinant identity, relating to an $n\times n$ matrix $\mathbb{A}$: First: $$ |\mathbb{A}||A_{r\,s,p\,q}| = |A_{r,p}||A_{s,q}| - |A_{r,q}|...
Honza's user avatar
  • 419
1 vote
1 answer
248 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 ...
Johnny T.'s user avatar
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6 votes
1 answer
190 views

Values of a pair of determinants

Let $\mathbf{x} = (x_0, x_1, x_2), \mathbf{y} = (y_0, y_1, y_2)$ be vectors over a field $\mathbb{F}$ of characteristic zero. Define the function $\displaystyle S(\mathbf{x}, \mathbf{y}) = x_2 (y_0^2 -...
Stanley Yao Xiao's user avatar
2 votes
0 answers
230 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 ...
GA316's user avatar
  • 1,219
2 votes
1 answer
386 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)...
GA316's user avatar
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3 votes
0 answers
111 views

Is the determinantal ideal of the span of a linearly independent set of rank-one matrices radical?

Let $k$ be an algebraically closed field, and let $X_1,\dots, X_n \in M_m(k)$ be rank-one matrices that are linearly independent over $k$. For a fixed integer $1 \leq r \leq m$, consider the ideal $I \...
Ben's user avatar
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22 votes
3 answers
1k views

On permanents and determinants of finite groups

$\DeclareMathOperator\perm{perm}$Let $G$ be a finite group. Define the determinant $\det(G)$ of $G$ as the determinant of the character table of $G$ over $\mathbb{C}$ and define the permanent $\perm(G)...
Mare's user avatar
  • 25.8k
6 votes
2 answers
316 views

Characteristic polynomial of checker matrix

For every integer $n > 0$, let $C_n$ be the $4n \times 4n$ matrix having $1$'s in all positions $(i, j)$ such that $i - j$ is even, $3$'s in the two diagonals determined by $|i - j| = 2n + 1$, and $...
MargeL's user avatar
  • 63
5 votes
2 answers
525 views

How to compute a more general version of Vandermonde / Cauchy double alternant determinant

Consider some variables $\{X_i\}_{1\le i \le n}$, $\{Y_i\}_{1\le i \le n}$, and $\{W_i\}_{1\le i \le n}$. Does anyone know how to compute the following determinant? $$ \det ~ \left(\frac{W_j^{i-1}}{...
Ahmadreza Momeni's user avatar
16 votes
2 answers
1k views

How to prove the determinant of a Hilbert-like matrix with parameter is non-zero

Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero? $$ \begin{pmatrix} \frac{1}{\...
Ahmadreza Momeni's user avatar
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 (...
Per Alexandersson's user avatar
7 votes
1 answer
282 views

Has vol. 3A of Cullis's "Matrices and Determinoids" been scanned and vol. 3B been archived?

This is a borderline question, but I'm going to risk posing it. Cuthbert Edmund Cullis (1875?-1955?) was a somewhat obscure British mathematician whose opus magnum was a multi-volume treatise called ...
darij grinberg's user avatar
9 votes
2 answers
464 views

Minors of low rank skew-symmetric matrix

Let $A$ be an $n\times n$ skew-symmetric matrix of rank $r$. Given subsets $X$ and $Y$ of row and column indices respectively, let $A_{X,Y}$ denote the submatrix of $A$ obtained by only keeping rows ...
Naysh's user avatar
  • 455
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 ...
Denis Serre's user avatar
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2 votes
0 answers
238 views

Characteristic polynomials of some special matrices

This is related to question Matrix-valued periodic Fibonacci polynomials. I want to find integer-valued matrices $x$ such that the Fibonacci polynomials $f_n(x)$, defined by the recursion $f_n(x)=...
Johann Cigler's user avatar
1 vote
0 answers
63 views

Linear algebra - For symmetric matrix X $\in S^n$, prove the $a^T X a$ = $\det X \det(X_{n-1})$ , where $a_i$ = $(-1)^i M_{in} $ [closed]

Suppose we have a symmetric matrix X$\in S^n$, and $X_k$ denotes the submatrix consists of first $k$ rows and columns of X. If $\det X < 0$, but $\det X_1, ..., \det X_{n-1} > 0$. Let $a_i=(-1)^...
RayyyyySun's user avatar
17 votes
1 answer
796 views

Determinantal identities for perfect complexes

Let $S$ be a noetherian scheme. Let $V,W$ be vector bundles on $S$. There is a canonical isomorphism of line bundles $$ {\rm det}(V\otimes W)\cong{\rm det}(V)^{\otimes{\rm rk}(W)}\otimes{\rm det}(W)^{\...
Damian Rössler's user avatar
0 votes
1 answer
89 views

Determinant of a block matrix with dissimilar elements [closed]

I am looking for an expression that convert calculation of given quadratic form into determinant of some block matrix. I see this in that form (that may be incorrect): $x^T A x = \begin{vmatrix} x^T ...
dtn's user avatar
  • 145
2 votes
0 answers
160 views

Naive generalization of determinant from matrices to higher rank tensors

Recall that using the Levi-Cevita symbol the determinant can be written as $$\operatorname{det} A=\frac{1}{n!}\epsilon_{i_1\dots i_n}\epsilon_{j_1\dots j_n}A_{i_1j_1}\dots A_{i_nj_n}$$ Some ...
Weather Report's user avatar
5 votes
0 answers
322 views

Determinantal formula for plane partitions of shifted shape

For $\lambda = (\lambda_1,\ldots,\lambda_{\ell})$ a partition, a (weak) plane partition of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with nonnegative integers such that entries ...
Sam Hopkins's user avatar
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1 vote
0 answers
127 views

Is this determinant well-known?

Let $n, k$ be any positve integers. I'm wondering if the following determinant is well known $$D_{n,k}= \begin{vmatrix}1^k& 2^k & 3^k&\cdots & n^k \\2^k & 3^k & 4^k &\...
Tomas's user avatar
  • 799
7 votes
1 answer
474 views

How can one define a kind of "determinant" on a reduced group $C^*$ algebra?

Let $A$ be a unital $C^*$-algebra which is equipped with a faithful trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following ...
Ali Taghavi's user avatar
6 votes
2 answers
616 views

How large a subset of $\mathbb{F}_q^d$ can determine all determinants?

Denote by $\mathbb{F}_q$ a finite field with $q$ elements. For $\mathcal{P}$ be arbitrary subset of $\mathbb{F}_q^d.$ We define the set $$S:= \left\{ \det([x_1,x_2,\dots,x_d]): x_1,x_2,\dots,x_d \in ...
user avatar
2 votes
2 answers
212 views

Cartan determinants of subsets

Let $n \geq 3$ be fixed. We associate to every subset $S \subseteq \{1,...,n-1 \}$ a number, which we call Cartan determinant of $S$ (see the end of this post for a representation theoretic background)...
Mare's user avatar
  • 25.8k
0 votes
1 answer
149 views

What are all the possibilities of $A$ s.t. $\det(A)=k$?

Suppose we have $A \in M_3(\Bbb N\cup\{0\})$ s.t. sum of the elements of each row is $k $ for some fixed $k\in \Bbb N\cup\{0\}$. What are all the possibilities of $A$ s.t. $\det(A)=k$? We can start ...
Ri-Li's user avatar
  • 103
14 votes
1 answer
2k views

Why does this matrix have zero determinant?

This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual ...
Hailong Dao's user avatar
  • 30.3k
4 votes
0 answers
179 views

Are mixed discriminants and hyper-determinants the same thing?

Premise Hello, I'm trying to learn more about hyperdeterminants, but as I'm sifting through the literature I'm finding more and more papers on Mixed discriminant and I find their definitions extremely ...
Luca Cappelletti's user avatar
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 ...
cws's user avatar
  • 131
2 votes
0 answers
45 views

The total Wronskian

Given a sequence of function, $$F=\{f_1(x,t),f_2(x,t),f_3(x,t),\cdots,f_m(x,t)\},$$ we define the total Wronski determinant of this set of functions as $$W(F)=\det\begin{vmatrix}F\\D_xF\\D_tF\\\vdots\\...
W. mu's user avatar
  • 201
1 vote
0 answers
88 views

When does a matrix with high rank have a minor with disjoint rows and columns and high rank?

This is a somewhat open-ended followup question to Does an antisymmetric matrix with high rank have a minor with disjoint rows and columns and high rank? and Does a non-singular matrix have a large ...
H A Helfgott's user avatar
  • 19.3k
1 vote
2 answers
183 views

Determinant diagonal blocks compound matrix [closed]

Good afternoon, I would like to prove the equation \begin{equation} \begin{vmatrix} b_{1,1}I_d & b_{1,2}I_d & \cdots & b_{1,r}I_d \\ b_{2,1}I_d & b_{2,2}I_d & \cdots & b_{2,r}...
Durzot's user avatar
  • 11
9 votes
0 answers
121 views

Cartan determinant of stable categories

Let $A$ be a finite dimensional algebra with finitely many indecomposable non-projective modules $M_1, M_2,...,M_n$. Let $a_{i,j}:=\dim(\underline{Hom_A}(M_j,M_i))$ (the dimension of the stable Homs ...
Mare's user avatar
  • 25.8k
1 vote
0 answers
75 views

Identifying a determinantal condition

Has the following condition already been studied and, if so, is there a known class of functions that satisfy it? Condition. For a fixed $n > 0$, all the $2 \times 2$ minors of the matrix $$ \...
Alessandro Pezzoni's user avatar
2 votes
0 answers
74 views

Can we approximate this matrix field with an invertible matrix field?

Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set $$\begin{equation*} A(x,y)=\left( \begin{array}{cc} x & -y \\ y & x \end{array} \right) \end{...
Asaf Shachar's user avatar
  • 6,611
10 votes
0 answers
367 views

Interpretation of determinants on commutative rings

In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map. This interpretation conceptually depends ...
user's user avatar
  • 323
2 votes
0 answers
83 views

Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs?

Planar graph permanent can be reduced to determinants and so statistics should be amenable. Pick a uniformly random bipartite planar graph $G$ with $n$ vertices of each color and choose new additional ...
Turbo's user avatar
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