# Tagged Questions

**1**

vote

**0**answers

70 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 ...

**2**

votes

**3**answers

234 views

### Solving a quadratic matrix equation with non-squared matrix

I was trying to solve the problem of finding the value of a non-squared matrix $T$ ($n \times m$) which solves
$$ T^T T = X$$
where $X$ is a symmetric and positive semidefinite $m \times m$ matrix, ...

**4**

votes

**0**answers

102 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

**3**answers

345 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 ...

**4**

votes

**0**answers

209 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

**3**answers

323 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

**1**answer

57 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

**3**answers

261 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

**1**answer

355 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

**1**answer

135 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

**1**answer

133 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

**1**answer

161 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

**0**answers

47 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

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**

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**0**answers

105 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

**1**answer

467 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 ...

**0**

votes

**0**answers

155 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**

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**0**answers

151 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 ...

**9**

votes

**2**answers

834 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

**1**answer

163 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

**1**answer

139 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

**1**answer

263 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

**3**answers

295 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

**1**answer

520 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

**3**answers

264 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

**2**answers

175 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

**1**answer

162 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

**2**answers

286 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

**1**answer

226 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**

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**0**answers

883 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

**1**answer

562 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

**1**answer

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 ...