All Questions
Tagged with quadratic-forms linear-algebra
84 questions
2
votes
0
answers
156
views
Optimization of quadratic form with band matrices
Let $A_1$ be the $N \times N$-matrix for which $a_{i,j} = 1$ for $i=j$ and 0 otherwise. Let $A_2$ be the matrix for which $a_{i,j}=1$ for $|i-j| \leq 1$ and 0 otherwise. Similarly define $A_3$ (which ...
1
vote
1
answer
152
views
Positive solutions to simultaneous real quadratic equations
I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as
$diag(x)Ax=1$
$x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...
- 54.2k
2
votes
1
answer
747
views
Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?
Let $Q \subset \mathbb{P}^4$ be a smooth three-dimensional quadric over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$) and let $F$ be the Fano variety of lines on $Q$.
In "Iskovskikh ...
- 66.3k
1
vote
0
answers
396
views
Bound of spectral radius of polynomial of a complex matrix
I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
- 163
0
votes
1
answer
118
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Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space
Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation.
I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...
- 38k
5
votes
0
answers
596
views
Literature on Exponential of a Quadratic Form
Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials
\begin{align}
f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\...
4
votes
1
answer
544
views
Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?
Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...
- 108k
2
votes
2
answers
262
views
on the determination of a quadratic form from its isotropy group in char. 2
So this question is a continuation of the following one
[1] On the determination of a quadratic form from its isotropy group
For some motivations and relevant backgrounds related to this question ...
CommunityBot
- 1
2
votes
0
answers
123
views
What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?
We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$
$$
F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2)
$$
on the vector space $V^{\mathbb{R}}:={\...
- 14.2k
4
votes
0
answers
390
views
Fully Homomorphic Error Correction?
Consider a field $F$. Suppose we have two vectors $a,b\in F^n$, and an invertible matrix $G\in F^{n\times n}$. Let $c\in F^n$ be the point-wise product of $a$ and $b$, that is, $c_i=a_ib_i$. Let $x=...
- 3,979
7
votes
3
answers
2k
views
Optimization problem on trace of rotated positive definite matrices
Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R \,\...
- 7,514
4
votes
0
answers
835
views
Hermitian forms over quaternion algebra
Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let $\sigma=Int(i)\...
- 41
0
votes
1
answer
261
views
Name for a Specific Type of Non-Symmetric Bilinear Form
Let $V$ be a finite dimensional vector space, with some choice of basis $\{e_i\}_{i \in I}$. With respect to an idempotent bijection $B:I \to I$, define a bilinear form by
$$
g = \sum_{i=1}^N \lambda_{...
- 13k
0
votes
3
answers
2k
views
rank of outer product
I would like to ask if you may know how to prove this claim or any theorem related:
Given 9 points (x,y,z) lie on unit sphere in 3 dimensional space such that any 4 points are not on the same plane. ...
CommunityBot
- 1
2
votes
1
answer
2k
views
Maximum dimension of an isotropic subspace in a quadratic space
i hope my question is not too trivial.
Let's suppose we have a vector space $V$ with a unimodular quadratic form $q$ of signature $(m,n)$.
My question is: which is the maximum dimension of an ...
- 3,035
1
vote
1
answer
396
views
Tangent space to positive oriented Grassmannians
Let $L$ be a real vector space of dimension 22 and $q$ a quadratic form on $L$ of signature $(3,19)$.
Let $V\subset L$ be a positive oriented subspace of dimension 2 and $G^{po}(2,L)$ be the ...
- 32.4k
0
votes
1
answer
311
views
Subspace generated by positive vectors
Hi everyone, first of all i must admit i'm very familiar with quadratic forms and positive subspaces, so i'm sorry if my question is too trivial. So, here's my problem:
Let $L$ be a real vector space ...
- 35.2k
0
votes
1
answer
775
views
Positive subspaces of quadratic forms
here's my question:
Let $V$ be a k-dimensional vector space over $\mathbb{R}$ and $q$ a quadratic form on $V$ of signature $(m,n)$ , $m+n=k$.
We have $W\subset V$ a positive (with respect to the ...
- 148k
1
vote
0
answers
120
views
Tensor product with $\mathbb{R}$ of an even unimodular lattice
Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$.
By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyperbolic plane.
Now ...
3
votes
0
answers
107
views
pavings and quadratic forms
Hi,
let $L$ be a lattice isomorphic to $\mathbb{Z}^r$ for some positive integer $r$ and $E=L\otimes \mathbb{R}$.
An integral paving in $E$ is a set $\Sigma$ of integral polytopes (the vertices are ...
- 31
8
votes
1
answer
2k
views
A spectral inequality for positive-definite matrices
Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues
$$
\lambda_1 \leq \cdots \leq \lambda_n ,
$$
is there a sharp upper bound for the product $\lambda_2 \cdots \...
CommunityBot
- 1
1
vote
0
answers
192
views
Non-negative Quadratic forms with Exterior Forms
Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{...
0
votes
0
answers
270
views
Solution Existence of a System of Complex Quadratic Equations
Consider $ {x_k}_{k = 1}^K \in \mathbb{C}^{N \times 1}$ a set of $K$ complex vector variables of length $N$. I am interested in finding the existence of a solution of the following
quadratic set of ...
- 342
1
vote
1
answer
177
views
if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic?
Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form
$q: \Pi \oplus \Pi \rightarrow R $
$\big( \alpha ,\beta \big) \longrightarrow \alpha \...
- 108k
2
votes
1
answer
210
views
Gram matrix modulo 4
Suppose we have a full rank, integer sublattice $L$ of the integer lattice $\mathbb Z^d$, where we fix the dimension $d$. Consider the Gram matrix $M$ of $L$, relative to some basis for $L$, and ...
- 25.7k
2
votes
3
answers
348
views
if Y-X is positive semi-definite, are the eigenvalues of Y bigger?
So $X$ and $Y$ are Hermitian matrices (or just symmetric real) of size $n$ by $n$ and suppose $Y\succeq X$, namely $Y-X$ is positive-semidefinite. Now write the eigenvalues of $Y$ as $\alpha_1\leq\...
- 28.6k
10
votes
1
answer
2k
views
Over which fields is the Sylvester law of inertia valid?
Short version:
Over which fields is the (appropriate version of the) "Sylvester law of inertia" valid?
Long version:
Let $V$ be a finite dimensional vector space over the field $\Bbbk$ of ...
- 23.3k
8
votes
3
answers
431
views
Characterising semi-definite positiveness on vectors with non-negative entries
My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...
- 52.3k
5
votes
1
answer
183
views
Identification of conformal classes of pos def quadratic forms on R^2 with unit ball
One of the lemmas at the foundation of Teichmuller theory is as follows. Let $Q(x,y)$ be a positive definite quadratic form. Then there exists unique $\lambda \in \mathbb{R}$ and $\mu \in \mathbb{C}$...
- 68.9k
2
votes
2
answers
468
views
Orthogonal transformations fixing a subspace (setwise)
Let $(V,Q)$ be a non-degenerate quadratic space of dimension $n$ over an algebraically closed field of characteristic zero. Let $W$ be a subspace of $V$ of dimension $m < \frac12 n$ which is ...
CommunityBot
- 1
3
votes
1
answer
250
views
action of SO(q)
Let $(V,q)$ be a non-degenerate quadratic space. Then we know that for any $d$ with $0 \leq 2d \leq \dim V$, the group $O(q)$ of isometries of $(V,q)$ acts transitively on the set of totally isotropic ...
- 22.6k
8
votes
0
answers
1k
views
roots of quadratic forms
This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
- 118k
5
votes
1
answer
745
views
Decide how many non-negative solutions a set of multivariate quadratic equations have
Given a set of multivariate, quadratic, non-homogeneous equations, is there a way to decide how many non-negative roots it have?
Some explanations:
All the coefficients are real numbers.
The number ...
- 8,501
22
votes
1
answer
13k
views
Non-diagonalizable complex symmetric matrix
This is a question in elementary linear algebra, though I hope it's not so trivial to be closed.
Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with ...
- 156k