Questions tagged [determinants]
Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.
536 questions
4
votes
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answer
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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 ...
- 53
10
votes
2
answers
847
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 ...
- 4,827
4
votes
1
answer
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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
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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} = \...
- 1,186
1
vote
1
answer
222
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
120
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 ...
- 121
1
vote
0
answers
104
views
Characteristic polynomial of block tridiagonal matrix
Suppose that I have an $nk \times nk$ matrix of the form
$$
T_n = \left[\begin{array}{cccccc}
A&B&&&&\\
B^T&A&B&&&\\
&B^T&A&B&&\\
&&\...
- 13.7k
46
votes
7
answers
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views
If $\det(M)=ab$ is it true that $M=AB$ with $\det(A)=a, \det(B)=b$?
For which (commutative) rings $R$ and dimensions $n$ is the following claim true (or false)?
Claim: For all $a,b$ any $n\times n$ matrix $M$ with coefficients in $R$ and $\det(M)=ab$, can be factored ...
- 5,825
8
votes
2
answers
405
views
Vandermonde-type closed form for products of $3\times3$ minors?
Question. The classical Vandermonde identity says
$$
\prod_{1\le i<j\le n}(t_i-t_j)=
\det\!\begin{pmatrix}
1 & \cdots & 1\\
t_1 & \cdots & t_n\\
\vdots & \ddots & \vdots\\
...
10
votes
0
answers
357
views
Is there an elementary proof for this identity involving a simplex, quadratic forms and determinants?
In a recent project we found a curious identity for simplices (Theorem 5.6).
Let $\Delta\subset\Bbb R^d$ be a $d$-simplex with facets $F_0,...,F_d$, $v_i\in\Bbb R^d$ the vertex opposite to $F_i$, $u_i\...
- 14.5k
1
vote
1
answer
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views
Exact form of eigenvalues of pentadiagonal Toeplitz matrices
The tridiagonal Toeplitz matrices
$$\begin{pmatrix}
a & b & & \\
c & \ddots & \ddots \\
& \ddots & \ddots & b \\
& & c ...
4
votes
1
answer
199
views
Is the complete polarization of the determinant non-negative on tuples of hermitian positive semidefinite matrices?
$\DeclareMathOperator{\tr}{tr}$
I will explain my question for $n = 3$. Assume first that $A$ is a complex $3 \times 3$ matrix. If the eigenvalues of $A$ are $\lambda_i$, for $i = 1, \dots, 3$, we ...
- 5,377
1
vote
1
answer
123
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Determinant of a exponential/vandermonde type matrix
Let $x,y\in\mathbb{R}^n_{+}$, and define $A=[a_{ij}]_{i,j=1}^n\in\mathbb{R}^{n\times n}$ such that $a_{ij}=e^{x_iy_j}$. Do we know a closed form for $\text{det}(A)$? I'm interested in this because of ...
- 103
5
votes
1
answer
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views
Characterization of a convex sum of determinants
A quantum information problem I have been thinking about comes down to a linear algebra question that I dare to ask here.
Given: Integers $N,P$, and a set of real positive coefficients $C_1,C_2,\ldots ...
- 203k
3
votes
0
answers
93
views
Deformation of Gram matrix: determinant and eigenvalues
I want to compare the determinant and the eigenvalues of two gram matrices obtaining by deforming the first one by positive definite matrices.
Let us consider a family $(x_i)_{1\leq i \leq m}$ of ...
- 31