All Questions
Tagged with linear-algebra determinants
239 questions
4
votes
0
answers
99
views
Volume interpretation of number of perfect matchings in bipartite planar graphs
Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...
- 13.9k
9
votes
3
answers
409
views
Determinant of a block matrix with many $-1$'s
For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...
- 13.4k
9
votes
1
answer
349
views
A binomial determinant formula: a new variant
In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to
$$\det_{1\le i,j\le n}\...
- 43.2k
7
votes
1
answer
248
views
The determinant of a $4\times4$ matrix associated to some specific polynomial as follow
Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...
- 1,997
2
votes
0
answers
340
views
Determinant of a rank r perturbation
In the following paper:
Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations
on page 79, Golub et al. have the following set of equations:
$f(\lambda) = \...
- 121
7
votes
1
answer
1k
views
Block matrices and their determinants
For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows:
(a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal ...
- 43.2k
2
votes
0
answers
184
views
Evaluate a curious determinant with Legendre symbol entries
Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. R. Chapman's conjecture on the exact value of the determinant of
$$C_p:=\left[\left(\frac{i-j}p\right)\right]_{0\le i,j\le (p-...
- 15.6k
7
votes
1
answer
486
views
A new determinant question for primes $p\equiv3\pmod4$
Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by my question http://mathoverflow.net/questions/310301, here I introduce the matrices $A^+_p$ and $A^-_p$ ...
- 15.6k
13
votes
0
answers
348
views
A determinant problem for primes $p\equiv 1\pmod4$
Let $p$ be an odd prime, and let $A_p$ denote the matrix
$$[a_{ij}]_{1\le i,j\le (p-1)/2},$$
where
$$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&...
- 15.6k
13
votes
3
answers
1k
views
Is $-\det\big[\big(\frac{i^2+j^2}p\big)\big]_{1\le i,j\le (p-1)/2}$ always a square for each prime $p\equiv 3\pmod 4$?
Let $p$ be an odd prime and let $S_p$ denote the determinant
$$\det\left[\left(\frac{i^2+j^2}p\right)\right]_{1\le i,j\le (p-1)/2}$$
with $(\frac{\cdot}p)$ the Legendre symbol. By Theorem 1.2 of my ...
- 15.6k
6
votes
0
answers
177
views
Determinant arising in a problem from probability
Consider the determinant:
$$\Delta:=
\left|\begin{array}{cccc}
A_{j_1} & A_{k_1} & A_{j_1}A_{k_1} & 1 \\
A_{j_2} & A_{k_2} & A_{j_2}A_{k_2} & 1 \\
A_{j_3 } & A_{k_3 } &...
- 783
1
vote
0
answers
268
views
Determinant and restriction of scalar
Let $E/F$ be a finite separable extension of fields, and $V$ a finite dimensional vector space over $E$. Let $T\in\operatorname{End}_EV$ be a linear operator on $V$, and let $\det(T)$ be its ...
- 833
4
votes
1
answer
296
views
Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?
I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
- 188
2
votes
0
answers
402
views
lower bounding the absolute value of a determinant
In a problem that I'm working on currently, the following question came up and I feel this should be fairly elementary, but I couldn't prove it myself/couldn't find a reference. Any pointers or ...
- 305
1
vote
0
answers
2k
views
Decomposition of Determinant of Sub-Matrices of a Matrix
Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means
$$
{\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ...
- 211
19
votes
2
answers
2k
views
How to prove positivity of determinant for these matrices?
Let $g(x) = e^x + e^{-x}$. For $x_1 < x_2 < \dots < x_n$ and $b_1 < b_2 < \dots < b_n$, I'd like to show that the determinant of the following matrix is positive, regardless of $n$:
...
- 325
6
votes
2
answers
738
views
Probability of a large random integer Matrix to have zero determinant
Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$
I would like to ...
- 355
9
votes
2
answers
763
views
Parallelepiped is defined by the volumes of its faces
Let $v_1,...,v_n\in \mathbb{R}^n$ be linearly independent. The parallelepiped defined by these vectors is $P(v_1,...,v_n)=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Observe that while the ...
- 5,529
13
votes
1
answer
625
views
A difficult determinant
(EDIT: I have removed the denominators I had in a previous version as they were superfluous)
The $N\times N$ determinant
$$D(a,\vec{b})=\det\left((2N+a+b_j-i-j)!\right)$$
has the nice form
$$D(a,\...
- 2,552
1
vote
0
answers
148
views
Traces in associative algebras
Are there some books or papers about the general definition of traces:
If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...
- 283
12
votes
2
answers
2k
views
Determinant of identity matrix plus Hilbert matrix
I am looking for the determinant
$$ \det(I_n + H_n) $$
where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by
$$ [H_n]_{ij} = \frac{...
- 121
3
votes
3
answers
979
views
Determinant of Jacobian and directional derivatives
I have a function $f: \Re^2 \to \Re^2$ and would like to understand why
$$|Jf(x)|=\max_\theta|D_\theta f(x)|\cdot\min_\theta|D_\theta f(x)|$$
that is, why the determinant of the Jacobian of $f$ at $...
- 39
8
votes
0
answers
492
views
Strange determinant inequality $\det(C+ xA) \det(C-xA) \le (\det C)^2$
Let $A$ be an all-one $3$-by-$3$ matrix, let $C$ be a $3$-by-$3$ matrix, and let $x$ be a real number. How might one prove the following inequality?
$$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
- 99
4
votes
2
answers
2k
views
Determinant of structurally symmetric $n$-banded matrix?
Is there a formula to compute the determinant of a structurally symmetric $n$-banded matrix? I am specifically interested in the 5-banded matrix:
$$ \left[\begin{matrix}
c_{0} & s_{0} & 0 &...
- 143
6
votes
1
answer
623
views
A curious determinant of quadratic forms
In a work about the Wave Equation, I encountered the following symmetric matrix of size $1+n$, whose entries are quadratic forms. The arguments are a scalar $a$ and a vector $X\in k^n$.
$$S(a,X)=\...
- 52.3k
7
votes
2
answers
2k
views
Determinant of a sub-matrix of the classical adjoint
Let $A$ be a square matrix of order $n$, say with complex coefficients, and let $M$ be the plain matrix of minors of $A$ of order $n-1$ (no transpose, no sing changes). Let $I$ and $J$ be $r$-...
- 60.5k
3
votes
0
answers
104
views
Rank relation to maximum subpermanent and subdeterminant?
Given a $\pm1$ matrix $M$ of rank $r$ let the largest subdeterminant be $d$ and let the largest subpermanent be $p$.
Are there relations/bounds that connect $r$, $d$ and $p$?
Are there geometric and ...
- 13.9k
-1
votes
1
answer
195
views
Determinant of $Z^TZ$ [closed]
If one is looking at the characteristic polynomial of the $m \times m$ dimensional matrix $Z^TZ$ then apparently the coefficient of $(-1)^{m-k}$ in it can be written as, $\sum_{U \subset [m], V \...
- 2,246
55
votes
0
answers
2k
views
What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?
Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the ...
- 1,238
8
votes
3
answers
2k
views
Optimization problem with determinant as objective
Let $A$ be a given symmetric positive definite $N\times N$ matrix. I need to find a symmetric positive semi-definite matrix $S$ which is the solution to the following optimization problem
\begin{align}...
- 1,421
0
votes
0
answers
283
views
A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular
A paper I'm reading in representation theory states the following result:
Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
- 6,180
0
votes
1
answer
553
views
Directed graph cycles and the inverse of a weighted adjacency matrix
Let us view a matrix $B \in \mathbb{R}^{n \times n}$ as the weighted adjacency matrix of a directed graph $G$, i.e. there is an edge $i \to j$ in $G$ if $B_{ij} \neq 0$. Assume further that
$B$ does ...
- 1,731
3
votes
0
answers
105
views
Can one do better than using general purpose determinant algorithms when using the Fisher-Kasteleyn-Temperley method for perfect matchings?
Questions.
(numerical.generalPfaffian) Is it proved anywhere that in general it is not easier0 to calculate the determinant (over $\mathbb{Q}$) of the skew-symmetric signed adjacency matrix defined ...
- 6,051
5
votes
0
answers
315
views
Is there a matrix with this specific quadratic determinant?
We have $\det M=(a+b)(c+d)$ where
$M=\begin{bmatrix}
a& 0& -1& 0\\
0& c& 0& -1\\
b& 0& 1& 0\\
0& d& 0& 1
\end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
- 13.9k
7
votes
3
answers
1k
views
Determinant of correlation matrix of autoregressive model
I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form
$$R = \begin{pmatrix}
1 & r & \cdots & r^{d-...
- 719
7
votes
1
answer
3k
views
When does the determinant distribute over addition?
When does $\det(A+B)=\det(A)+\det(B)$ hold?
I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.
- 13.9k
8
votes
1
answer
726
views
A direct proof of a property of symmetric 2x2-determinants
Let $f(a,b,c)=\det\begin{pmatrix}a &b\\ b& c\end{pmatrix}\in\mathbb{R}[a,b,c]$ be the determinant of a $2 \times 2$ real symmetric matrix.
Let $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ ...
- 14k
5
votes
2
answers
635
views
Some curious Hankel determinants
Let $f(n,q)=\prod_{j=1}^na(q^j)$ for a polynomial $a(q)$ and let $d(n)=\det(f(i+j,q))_{i,j=0}^n$ be its Hankel determinant.
Computer experiments suggest that
$$\lim_{q\to1}\frac{d(n)}{(q-1)^\binom{n+...
- 5,622
15
votes
1
answer
8k
views
On the determinant of a class symmetric matrices
Consider the matrix $2\times2$ symmetric matrix:
$$
A_2=\begin{pmatrix} 1 & a_1 \\ a_1 & 1\end{pmatrix}.
$$
It's clear that the restriction $|a_1|<1$ implies that $\det(A_2)>0$. Moreover,...
- 563
4
votes
2
answers
239
views
Distribution of $0$-$1$ matrices
Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough.
What does the ...
- 309
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
12
votes
1
answer
624
views
Determinants: periodic entries $0,1,2,3$
Consider an $n\times n$ matrix $M_n$ where the sequence
$$\{1,2,3,\dots,n^2\} \mod 4=\{1,2,3,0,1,2,3,\dots\}$$ forms a clock-wise spiral, in that given order. For example,
$$M_4=\begin{bmatrix} 1&...
- 43.2k
14
votes
2
answers
873
views
"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
- 43.2k
8
votes
1
answer
321
views
"Almost Hankelized" numerical Vandermonde
One of the more utilized determinant is that of Vandermonde's
$$\begin{vmatrix}
1&x_1&x_1^2&\dots&x_1^{n-1}\\
1&x_2&x_2^2&\dots&x_2^{n-1}\\
\ldots&\ldots&\...
- 43.2k
5
votes
2
answers
335
views
Determinant of the "quantum" version of the group $\mathbb{Z}_n$
Let $[0]_q:=0$ and $[n]_q:=\frac{1-q^n}{1-q}=1+q+\cdots+q^{n-1}$, for $n\geq1$.
Question. Is there a closed formula (with proof) for the determinant of the matrix of $(i,j)$-entries
$$[i+j\bmod n]...
- 43.2k
2
votes
1
answer
325
views
Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
- 13.4k
4
votes
1
answer
183
views
dyadically recursive matrices: Part II
We present some variation on this MO question which is equally amusing to me.
Define the $2^{n-1}\times 2^{n-1}$ matrix $A_n$ recursively as follows: $A_1(a_1)=\begin{pmatrix} a_1\end{pmatrix}$ and
$$...
- 43.2k
5
votes
2
answers
203
views
dyadically recursive matrices: Part I
Introduce the $2^{n-1}\times 2^{n-1}$ matrix $B_n$ recursively as follows: $B_1(b_1)=\begin{pmatrix} b_1\end{pmatrix}$ and
$$B_n(b_1,\dots,b_n)=\begin{pmatrix} B_{n-1}(b_1,\dots,b_{n-1})& b_nJ_{n-...
- 43.2k
11
votes
3
answers
918
views
yet another determinant and inverse of a matrix
This problem is some variation of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c &...
- 43.2k
1
vote
0
answers
143
views
How to show the determinant of $B - I$ is zero? [closed]
Let $n \geq 2$ be a positive integer and
$$\beta_i= \left(
\begin{array}{c|c c|c}
I_{i-1} & 0 & 0 & 0\\
\hline
0 & 1-q & q & 0\\
0 & 1 & 0 & 0\\
\hline
...
- 113