Questions tagged [determinants]

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

Filter by
Sorted by
Tagged with
6
votes
2answers
223 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 $...
5
votes
1answer
270 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}}{...
16
votes
2answers
986 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}{\...
1
vote
0answers
202 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 (...
7
votes
1answer
235 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Has vol. 3A of Cullis's “Matrices and Determinoids” been scanned and vol. 3B been archived?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 ...
6
votes
1answer
95 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 ...
13
votes
1answer
346 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 ...
2
votes
0answers
210 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)=...
1
vote
0answers
54 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)^...
11
votes
0answers
280 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)^{\...
0
votes
1answer
39 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 ...
0
votes
0answers
26 views

Name for product of square-free content of invariant factors of a matrix

Let $M(\lambda)$ be a possibly non-square polynomial matrix (over $\mathbb{R}$ or $\mathbb{C}$ is sufficient for me, but could be more general). By standard theory, it can be put into Smith normal ...
2
votes
0answers
79 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 ...
5
votes
0answers
159 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 ...
1
vote
0answers
115 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 &\...
4
votes
1answer
275 views

How can one define a kind of “Determinant” on reduced group $C^*$ algebra?

Let $A$ be a unital $C^*$ algebra which is equipped with a faithfull trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following ...
6
votes
2answers
409 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 ...
2
votes
2answers
174 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)...
0
votes
1answer
141 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 ...
13
votes
1answer
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 ...
4
votes
0answers
73 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 ...
2
votes
0answers
106 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 ...
0
votes
0answers
48 views

Any trace-determinant inequalities for P/P0-matrices?

Two very purpolar inequalities involving trace and determinant are the following: If $A$ is psd of size $n$, then $\frac{1}{n}\text{tr}(A) \ge \det (A)^{1/n}$. If $A$ is psd, then $\log\det(A) \le \...
2
votes
0answers
28 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\\...
1
vote
0answers
83 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 ...
1
vote
2answers
72 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}...
9
votes
0answers
98 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 ...
1
vote
0answers
69 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 $$ \...
2
votes
0answers
69 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{...
10
votes
0answers
245 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 ...
2
votes
0answers
49 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 ...
0
votes
1answer
137 views

Positive definite matrix

We have $a_1,a_2,...,a_n\in (0,1)$ and matrix M= \begin{bmatrix}2a_1&a_2&a_3&.&.\\a_2&2a_2&a_3&.&.\\a_3&a_3&2a_3&.&.\\.&.&.&.&.\end{...
9
votes
2answers
380 views

Matrix of cosecants appearing in equivariant index computations

In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8....
8
votes
1answer
425 views

Property of the trace on finitely generated projective modules

Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $...
4
votes
0answers
104 views

Positivity of q-analogs of central binomial coefficients?

With the usual $q-$notations $[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q},$ $[n]_q!=[1]_q[2]_q\cdots[n]_q$ and $\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$ let $$b(n,k,r,q)=\det\left(q^{r\...
3
votes
1answer
308 views

Determinant involving traceless unitary hermitian matrices

Let $S$ be the set of complex $N\times N$ matrices that are traceless, unitary and hermitian. A friend asked me the following question, motivated by a problem in condensed matter physics: Is it ...
5
votes
0answers
176 views

On the determinants $\det\left[(i\pm j)\left(\frac{i\pm j}p\right)\right]_{1\le i,j\le(p-1)/2}$

Let $p$ be an odd prime and define $$D_p^+:=\det\left[(i+j)\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}$$ and $$D_p^{-}:=\det\left[(i-j)\left(\frac{i-j}p\right)\right]_{1\le i,j\le(p-1)/2},$$ ...
7
votes
1answer
439 views

On a problem for determinants associated to Cartan matrices of certain algebras

This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
11
votes
0answers
254 views

More mysterious properties of Gram matrix

This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question. The following fact could be extracted from 0402087: For any $a_i\...
3
votes
1answer
208 views

Determinant of an “almost cyclic” matrix

Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let $\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\...
10
votes
0answers
212 views

Generalized eigen property of a matrix

Given a $n \times n$ invertible matrix $A$, I am interested in the set $$ \mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}. $$ Thus, for all eigenvalues $\lambda_i$, we have $...
5
votes
0answers
120 views

Conjecture for a certain Cauchy-type determinant

Given the Cauchy-like matrix $$ \mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{ \Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right) }{ \Gamma(m)\,\Gamma(n) } \frac{m-\frac{3}{4}} {\...
1
vote
0answers
126 views

On $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le p-1}$ and $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le (p-1)/2}$

QUESTION. Is my following conjecture true? If true, how to prove it? Conjecture. Let $p$ be a prime with $p\equiv5\pmod 6$, and define the matrices $A_p$ and $B_p$ by $$A_p:=\left[\frac1{i^2-ij+j^2}\...
2
votes
0answers
134 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 $P$-...
11
votes
2answers
515 views

Classification of algebras of finite global dimension via determinants of certain 0-1-matrices

I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background). A Morita-...
2
votes
2answers
227 views

Equal-valued determinants in search of a proof: Part III

Encouraged by David's proof for my earlier MO question, let's consider a similar problem. I can prove the below equality by computing each of the two sides, directly. That means, there is an ...
11
votes
1answer
457 views

Catalan determinants in search of a proof: Part II

This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof. ...
7
votes
1answer
275 views

Deligne Pairing v.s. Weil Pairing on a Family of curves

We have the Deligne Pairing on a family of curve $\pi:X\to S$ by using $$\langle L,M\rangle_{\mathrm{Pic}^0(X/S)}=\det R\pi_*(L\otimes M) \otimes (\det R\pi_*L)^{-1}\otimes (\det R\pi_*M)^{-1} \otimes ...
11
votes
2answers
895 views

Determinantal symmetry: proof requested: Part I

Consider the determinantal function $$F(a,b,c):=\det\left[\binom{i+j+a+b}{i+a}\right]_{i,j=0}^{c-1}.$$ I would like to ask: QUESTION. Can you provide an argument, combinatorial or otherwise, to ...
3
votes
1answer
385 views

Coordinate free expression for the determinant of a $2 \times 2$ matrix

Let $E\rightarrow X$ be a rank two vector bundle on a variety $X$. How can one write an abstract map $Sym^2(E)\rightarrow (\wedge^2 E)^{\otimes 2}$ that in local coordinates, when we fix a frame of $E$...

1
2 3 4 5
8