Questions tagged [determinants]
Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.
499
questions
3
votes
2
answers
217
views
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_{...
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 ...
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.
$$\...
2
votes
0
answers
323
views
For the following class of matrices, are the determinants invariant under permutations?
I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
2
votes
1
answer
247
views
Some determinants which are closely related to recurrences
Let the sequence $(a(n,k))_{ n \in \mathbb{Z}}$ satisfy $$\sum_{j=0}^k c(k,j)a(n-j,k)=0$$ with $c(k,j)=c(k,k-j)$ and $c(k,0)=1$ and with initial values $a(-n,k)=0$ for $1\leq n\leq{k-1}$ and $a(0,k)=...
1
vote
0
answers
108
views
Is there a bipartite graph whose determinant corresponds to number of perfect matchings?
Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$...
5
votes
1
answer
122
views
Identity relating iterated determinant line bundles
Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
2
votes
0
answers
128
views
Pfaffian generalization
The identity
$$\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|=\...
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}^{...
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 $n\times m$ ...
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 ...
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}
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 ...
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 &...
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:
$$
|\mathbb{A}||A_{r\,s,p\,q}| = |A_{r,p}||A_{s,q}| - |A_{r,q}|...
1
vote
1
answer
248
views
When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?
Let $A$ be an $n \times n$ real symmetric matrix.
Let
$$
M = \begin{pmatrix} A & X \\ X^T & A \end{pmatrix}
$$
where $X$ is a real invertible $n \times n$ matrix. I am interested in finding ...
6
votes
1
answer
190
views
Values of a pair of determinants
Let $\mathbf{x} = (x_0, x_1, x_2), \mathbf{y} = (y_0, y_1, y_2)$ be vectors over a field $\mathbb{F}$ of characteristic zero. Define the function
$\displaystyle S(\mathbf{x}, \mathbf{y}) = x_2 (y_0^2 -...
2
votes
0
answers
230
views
Roots of determinant of matrix with polynomial entries — a generalization
For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...
2
votes
1
answer
386
views
Roots of determinant of matrix with polynomial entries
Let $p_1, p_2,\dots, p_n$ and $q_1,q_2,\dots,q_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying
$$a_{ij} = \begin{cases} p_i(x) & \text{ if } i = j, \\ q_i(x)...
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 \...
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)...
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 $...
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?
$$
\det ~ \left(\frac{W_j^{i-1}}{...
16
votes
2
answers
1k
views
How to prove the determinant of a Hilbert-like matrix with parameter is non-zero
Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero?
$$
\begin{pmatrix}
\frac{1}{\...
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 (...
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 ...
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 ...
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 ...
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)=...
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)^...
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
$$
{\rm det}(V\otimes W)\cong{\rm det}(V)^{\otimes{\rm rk}(W)}\otimes{\rm det}(W)^{\...
0
votes
1
answer
89
views
Determinant of a block matrix with dissimilar elements [closed]
I am looking for an expression that convert calculation of given quadratic form into determinant of some block matrix. I see this in that form (that may be incorrect):
$x^T A x = \begin{vmatrix} x^T ...
2
votes
0
answers
160
views
Naive generalization of determinant from matrices to higher rank tensors
Recall that using the Levi-Cevita symbol the determinant can be written as
$$\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}$$
Some ...
5
votes
0
answers
322
views
Determinantal formula for plane partitions of shifted shape
For $\lambda = (\lambda_1,\ldots,\lambda_{\ell})$ a partition, a (weak) plane partition of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with nonnegative integers such that entries ...
1
vote
0
answers
127
views
Is this determinant well-known?
Let $n, k$ be any positve integers. I'm wondering if the following determinant is well known
$$D_{n,k}= \begin{vmatrix}1^k& 2^k & 3^k&\cdots & n^k \\2^k & 3^k & 4^k &\...
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 ...
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
$$S:= \left\{ \det([x_1,x_2,\dots,x_d]): x_1,x_2,\dots,x_d \in ...
2
votes
2
answers
212
views
Cartan determinants of subsets
Let $n \geq 3$ be fixed.
We associate to every subset $S \subseteq \{1,...,n-1 \}$ a number, which we call Cartan determinant of $S$ (see the end of this post for a representation theoretic background)...
0
votes
1
answer
149
views
What are all the possibilities of $A$ s.t. $\det(A)=k$?
Suppose we have $A \in M_3(\Bbb N\cup\{0\})$ s.t. sum of the elements of each row is $k $ for some fixed $k\in \Bbb N\cup\{0\}$. What are all the possibilities of $A$ s.t. $\det(A)=k$?
We can start ...
14
votes
1
answer
2k
views
Why does this matrix have zero determinant?
This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual ...
4
votes
0
answers
179
views
Are mixed discriminants and hyper-determinants the same thing?
Premise
Hello, I'm trying to learn more about hyperdeterminants, but as I'm sifting through the literature I'm finding more and more papers on Mixed discriminant and I find their definitions extremely ...
3
votes
0
answers
133
views
Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)
For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
2
votes
0
answers
45
views
The total Wronskian
Given a sequence of function, $$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\\...
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 ...
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}...
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 ...
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
$$
\...
2
votes
0
answers
74
views
Can we approximate this matrix field with an invertible matrix field?
Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set
$$\begin{equation*}
A(x,y)=\left(
\begin{array}{cc}
x & -y \\
y & x
\end{array} \right)
\end{...
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 ...
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 ...