Questions tagged [determinants]
Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.
543 questions
7
votes
1
answer
522
views
A basis-free formula for the determinant as a polynomial
Let $V$ be an $n$-dimensional $\mathbb{K}$-vector space. By a simple calculus trick (*) on homogeneous functions of degree $n$ the determinant is a linear map on the $n$-th symmetric power of the ...
- 1,248
7
votes
1
answer
415
views
Multidimensional Glasser's master theorem
Let $f=(f_1,\dots,f_n)$, with
$$
f_i(x_1,\dots,x_n)=a_{ii}T_i(x_i)+\sum_{j\not=i}a_{ij}x_j
$$ where
$$
T_i(x_i)=x_i-d_i-\sum_{l}\frac{|b_{i,l}|}{x_i-c_{i,l}}.
$$
and $A=(a_{ij})$ is positive-definite....
- 173
1
vote
0
answers
187
views
Hadamard determinant problem and the special orthogonal group
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PL{PL}$I was trying to reduce the Hadamard problem of calculating the maximum value of the determinant of a $\{1,-1\}$-matrix to the problem of ...
- 401
13
votes
2
answers
447
views
Determinant of Hessenberg matrix with $1/n^2$ entries
Background
Consider the $(n \times n)$ Hessenberg matrix
$$ A_{n} := \begin{pmatrix} 1/2 & 1/3 & 1/4 & 1/5 & \dots & \dots & 1/(n+1) & \dots \\
...
- 5,387
9
votes
2
answers
382
views
Representing homogeneous forms as a determinant?
Let $F(x_1, ...., x_n)$ be a homogeneous form degree $d$ defined over $\mathbb{Q}$.
I am interested in learning about what is known about expressing
$$
F(x) = \det M
$$
where $M$ is a matrix whose ...
- 3,889
10
votes
0
answers
177
views
Does the Grassmannian scheme have meet and join?
The case of a field
Let $\mathbf{k}$ be a field (for now). Let $n\in\mathbb{N}$, and set $\left[
n\right] :=\left\{ 1,2,\ldots,n\right\} $. Consider the $\mathbf{k}$-vector
space $V:=\mathbf{k}^{n}$...
- 35.9k
7
votes
0
answers
290
views
Disjoint nonsingular matrix partition
Determine all $n\times n$ matrices $A$ with the following property
For any square submatrix $B$ of $A$, $\det(B)\det(\bar B)=0$. Here, $\bar B$ is the complementary matrix of $B$ obtained by deleting ...
- 467
4
votes
1
answer
249
views
Cayley transform - determinant/permanent faster computation
Let $A$ be a $0/1$ matrix in $\mathbb Z^{n\times n}$ such that $I+A$ is invertible $\bmod 3$. Consider $Q=(I-A)(I+A)^{-1}$.
Let $Det(M)$ and $Per(M)$ be determinant and permanent respectively of ...
- 199
11
votes
2
answers
891
views
Linear algebraic lemma in Weil II
The following linear algebraic lemma is used in Weil II to prove properties of $\tau$-real sheaves:
Lemma Let $A$ be a complex matrix and let $\overline A$ be its complex conjugate. Then the ...
- 5,236
4
votes
1
answer
239
views
A system of T-functions on $(0,\infty)$
Let $r>1$ be real and
\begin{align*}
f_1(x) &= 1,\\
f_2(x) &= (x+4)^r,\\
f_3(x) &=(x+4)^r(x+3)^r,\\
f_4(x) &= (x+4)^r(x+3)^r(x+2)^r,\\
f_5(x) &=(x+4)^r(x+3)^r(x+2)^r(x+1)^r.
\...
- 258
2
votes
0
answers
128
views
Determinant related to a union-closed family of sets
Let $\mathcal{F} = \{A_1, \ldots, A_n\}$ be a union-closed family of sets with universe $\bigcup_{i=1}^n A_i = [q] = \{1, \ldots, q\}$.
Let $M = [m_{ij}]$ be the $n \times n$ matrix with $m_{ij} = \...
- 1,196
1
vote
1
answer
243
views
Positive roots of real exponents function
Let $$f(x) = a+b(x+3)^r+c(x+3)^r(x+2)^r+d(x+3)^r(x+2)^r(x+1)^r,$$ where $r>1$ and $a,b,c,d$ are real numbers. I need to prove that $f(x)$ can't have more than $3$ positive zeros.
Attempts -
By ...
- 258
2
votes
0
answers
125
views
Is there a 'determinant' of a two-variable function when treated as a linear map?
A two variable real function $F(y,x)$, defined on $[c,d] \times [a,b]$, can be thought of as a linear map between functions of different domains by:
$$
g(y) = \int^a_bF(y,x)f(x)dx
$$
This has very ...
- 169
1
vote
0
answers
118
views
Characteristic polynomial of block tridiagonal matrix
Suppose that I have an $nk \times nk$ matrix of the form
$$
T_n = \left[\begin{array}{cccccc}
A&B&&&&\\
B^T&A&B&&&\\
&B^T&A&B&&\\
&&\...
- 13.8k
61
votes
7
answers
3k
views
If $\det(M)=ab$ is it true that $M=AB$ with $\det(A)=a, \det(B)=b$?
For which (commutative) rings $R$ and dimensions $n$ is the following claim true (or false)?
Claim: For all $a,b$ any $n\times n$ matrix $M$ with coefficients in $R$ and $\det(M)=ab$, can be factored ...
- 6,025