4
votes
0answers
85 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 ...
5
votes
3answers
290 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 ...
5
votes
0answers
176 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 ...
3
votes
3answers
279 views

A NICE necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure!

Let $$ A = \begin{pmatrix} \sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\ -a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\ \vdots & \vdots & \ddots & ...
0
votes
1answer
54 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 ...
0
votes
3answers
253 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. ...
0
votes
1answer
242 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 ...
0
votes
1answer
126 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 ...
1
vote
1answer
132 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 ...
0
votes
1answer
149 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 ...
1
vote
0answers
39 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
0answers
85 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 ...
2
votes
0answers
97 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} ...
7
votes
1answer
423 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 ...
1
vote
0answers
74 views

Orthogonal trasformations with trivial spinor norm as product of reflections $r_w$ with $(w,w)=-2$

I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard i mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element of $O^+(\Lambda)$ ...
0
votes
0answers
142 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 ...
1
vote
0answers
147 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 ...
8
votes
2answers
690 views

On the Positive Definiteness of a Linear Combination of Matrices

In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated. QUESTION: Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
1
vote
1answer
162 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 ...
2
votes
1answer
133 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 ...
3
votes
1answer
261 views

Can an ellipsoid be moved freely inside another ellipsoid?

An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid equivalent if one can be ...
2
votes
3answers
292 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 ...
6
votes
1answer
478 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 ...
7
votes
3answers
259 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 ...
2
votes
1answer
150 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 ...
5
votes
1answer
160 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 ...
2
votes
2answers
283 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 ...
3
votes
1answer
225 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 ...
8
votes
0answers
860 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 ...
5
votes
1answer
561 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 ...
5
votes
1answer
3k 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 ...