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4 votes
1 answer
236 views

If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?

Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
1 vote
1 answer
133 views

Graceful labeling of the complete bipartite graph and its laplacian quadratic form diagonalized

A graceful labeling of a connected simple undirected graph $G=(V,E)$ is a map $f:V\to\lbrace 1,...,|E|+1\rbrace$ such that for all $t\in\lbrace 1,...,|E|\rbrace$ there is a (trivially unique) $\langle ...
1 vote
1 answer
153 views

How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?

Scenario I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$. Problem I know that ${x}$ ...
0 votes
0 answers
93 views

Orthogonalization of symmetric non-degenerate bilinear forms

It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
2 votes
2 answers
1k views

Rank of a linear combination of quadratic forms

Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...
0 votes
0 answers
111 views

Totally isotropic space for bilinear pairing over ring

A duplicate of this: Consider the following well-known inequality: Let $b$ be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$ a totally ...
2 votes
0 answers
55 views

Quadratic surjective map between spheres

The quadratic function $f:\mathbb R^4\to\mathbb R^3$ $$f(a,b,c,d)=\begin{bmatrix} 2(ac + bd)&2(ad - bc)&a^2 + b^2 - c^2 - d^2\end{bmatrix}$$ surjectively maps the sphere $S^3$ to the sphere $S^...
15 votes
2 answers
1k views

Positive quadratic polynomial

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$. Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$. Is it possible to find a polynomial $\tilde q$ ...
12 votes
1 answer
902 views

Positive 4-form

Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$. Let $Q$ be a quadratic form on $W$. Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
4 votes
0 answers
216 views

Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism

Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
2 votes
0 answers
266 views

Solutions to the quadratic matrix equation $X A X^T = B$

Let $A, B \in \mathbb{R}^{n \times n}$ be symmetric, positive-semidefinite, full-rank matrices. I would like to understand the set of $X \in \mathbb{R}^{n \times n}$ which are themselves symmetric and ...
4 votes
1 answer
277 views

Is there a good notion of kernels of quadratic forms on abelian groups?

Let $G$ be an abelian group and let $q:G \to \mathbb{Q/Z}$ be a quadratic form, i.e. $q(a)=q(-a)$ and $b(x,y)=q(x+y)-q(x)-q(y)$ is a bihomomorphism. On vector spaces, when people speak about the ...
1 vote
1 answer
70 views

A question about the sign of quadratic forms on nonnegative vectors

Let $M$ be a real square matrix of order $n\ge 3$. Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$. Can ...
1 vote
1 answer
298 views

Degenerate representation

Let $\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$ be a faithful representation of the commutative Lie algebra $\mathbb{R}^n$ into the Lie algebra of skew-symmetric matrices. There is an orthonormal basis ...
1 vote
0 answers
87 views

Continuous choice of null directions for a family of bilinear forms

Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...
1 vote
0 answers
94 views

Maximal $l^2$ restriction of psd quadratic form on $\mathbb{R}^n$

I've been stuck on this problem for a few days as part of a bigger research problem and am looking for some help. Let $S : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a positive ...
0 votes
0 answers
92 views

Positive definite matrix and Hörmander theory

Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$ Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set $$ \varphi_{\...
2 votes
2 answers
323 views

Determinants of minors occurring 'within' determinant of full matrix

$A= (a_{ij})$ is an $n\times n$ symmetric positive matrix. It induces a quadratic form $f(x):= x^tAx$ on $\mathbb{R}^n$. $D_m$ denotes the determinant of the top left $m\times m$ submatrix of $A$ (or ...
0 votes
1 answer
199 views

Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $

Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
1 vote
0 answers
52 views

When the summands of a positive definite matrix are positive definite

Let $A,B$ be two real symmetric matrices. Let $C = A+B$ be a positive-definite matrix. Write $C>0$ for $C$ being positive-definite. Suppose that $A>0 \implies C>0$ and $B > 0 \implies C>...
1 vote
1 answer
263 views

A question on vectors in $\mathbb{R}^4$

Let $M$ be an invertible $4 \times 4$ real matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^4$. Consider the matrix $$\displaystyle \mathcal{M} = \mathcal{M}(\mathbf{u}, \mathbf{v}) = [\mathbf{u}...
5 votes
1 answer
342 views

Is the map $GL_n(\mathbb{Z})\to GL_n(\mathbb{Z}/2\mathbb{Z})$ surjective?

Suppose $F$ is a field. I want to know whether the map $GL_n(GW(F))\to GL_n(W(F))$ is surjective, where $GW$ means Grothendieck-Witt and $W$ means Witt. In the case $F$ is algebraic closed, it reduces ...
12 votes
2 answers
3k 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, ...
5 votes
0 answers
237 views

Linearly independent quadratic forms vanishing on a finite set of points

The question I am interested in can be summed up as follows: given positive integers $n,m,k$, how do we write down $m$ linearly independent quadratic forms $Q_1, \cdots, Q_m \in \mathbb{C}[x_0, \cdots,...
3 votes
0 answers
138 views

Is there a method to solve a non-linear quadratic matrix equation?

I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$ Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...
11 votes
0 answers
158 views

Characterization of certain 4-dimensional lattices

Let $\Lambda \subset {\bf Q}^4$ be a lattice, i.e., $\Lambda$ is a free abelian group and $\Lambda \otimes {\bf Q} = {\bf Q^4}$. The determinants of those dilation-rotations (i.e. linear maps of ${\bf ...
2 votes
0 answers
114 views

Projective group of Plucker quadric over the reals

A somewhat elementary question but seemingly difficult to find a suitable reference: Consider the six-dimensional real space $\wedge^2(\mathbb R^4)$ with basis $e_i \wedge e_j \ (i < j)$ where $...
3 votes
3 answers
590 views

Solving diagonal simultaneous quadratic equations

A problem I am trying to solve has led to me to the following system of equations: $$A(x^2) + Bx + c = 0$$ Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...
2 votes
0 answers
111 views

Inseparable field extensions of degree p and linear independence

Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$. Is it true that the elements $1,(x-...
2 votes
0 answers
47 views

Generalization of positive definiteness for a non-smooth quadratic form

I have a "quadratic form" defined as follows: $$f(\vec x) = \sum_{i<j} c_{ij} x_i x_j + \sum_i a_i(x_i) x_i^2$$ where the coefficients $c_{ij}$ are fixed constants, but the coefficients $a_i(x_i)$...
5 votes
3 answers
3k 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 & \...
5 votes
1 answer
2k views

The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices? [closed]

Problem: Suppose we have two real, symmetric and positive definite square matrices $A$ and $B$, i.e., $$A_{ij}, B_{ij}\in \mathbb{R}$$ $$A^T=A$$ $$B^T=B$$ $$x^TAx>0 \forall x$$ $$x^TBx>0 \...
2 votes
2 answers
119 views

Correlation between the first and a random position of an ergodic bit sequence

Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme. Probabilistic version. Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...
1 vote
0 answers
53 views

How explicitly write a projective transformation between the conics over the univariate function field?

Consider the quadratic forms $$ Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2 $$ over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...
0 votes
0 answers
58 views

Quadrics over the univariate function field with discriminant of minimal degree

Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...
1 vote
0 answers
111 views

Reference request: Number of elliptic and hyperbolic quadratic forms of a given rank over a finite field

My question is over the finite field $\mathbf{F}_q$ of $q$ elements. It is well known that a symmetric matrix of odd rank corresponds to a parabolic quadratic form but even rank symmetric matrices ...
2 votes
0 answers
108 views

Angle between subspaces in indefinite signature

I know there is a notion of principal angles between subpaces of a Euclidean space, but what about subspaces of a finite dimensional vector space $V$ equipped with a non-degenerate quadratic form of ...
1 vote
0 answers
69 views

Question about Notation for Spaces of $n$-ary $k$-ic Forms

Define an $n$-ary $k$-ic form to be a polynomial over the integers of homogeneous degree $k$ in $n$ variables. In Section 1 of the paper "Higher Composition Laws I" (linked below), Bhargava writes $(\...
0 votes
1 answer
215 views

Vanishing bilinear forms

For a symmetric or antisymmetric bilinear form $\varphi$ on a vector space $V$, if $\varphi(x,y)=0$ then also $\varphi(y,x)=0$ ($x,y\in V$). I was wondering if this is also a necessary condition for ...
4 votes
3 answers
1k views

Solving a quadratic matrix equation with fat matrix

I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves $$T^T T = X$$ where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix. I saw this post, but ...
3 votes
1 answer
600 views

« Generalized simultaneous diagonalization » of a pair of symmetric, non-commuting, positive semi-definite matrices

I hope my question is trivial for some of you but for the time being I’m lost somewhere between the generalized eigenproblem, simultaneous diagonalization of quadratic forms, simultaneous SVD, ...
4 votes
1 answer
393 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 ...
10 votes
2 answers
2k views

When is a bilinear form equivalent to a trace form?

Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$ Now, given a ...
20 votes
3 answers
1k views

Simultaneous "orthonormalization" in $\mathbb{C}^4$

Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix. So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
6 votes
1 answer
623 views

A curious determinant of quadratic forms

In a work about the Wave Equation, I encountered the following symmetric matrix of size $1+n$, whose entries are quadratic forms. The arguments are a scalar $a$ and a vector $X\in k^n$. $$S(a,X)=\...
0 votes
0 answers
283 views

A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular

A paper I'm reading in representation theory states the following result: Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
3 votes
2 answers
1k views

Roots of quadratic vector equation

Given $$A_{i j k}X_j X_k + B_{ij} X_j + C_i = 0$$ where $A_{ijk}$, $B_{ij}$, and $C_i$ are arbitrary real numbers for all $i$, $j$, $k$ which are $N$-dimensional indices, such that $A_{ijk}=A_{ikj}$ ...
1 vote
1 answer
129 views

Redistribute diagonal entries of a matrix

Let $d = (d_1,...,d_k)^t$ with positive entries. Denote $D:=diag(d)$ and let $m > k$. What are sufficient conditions on $d$ and $m$ so that there exists $V \in \mathbb{R}^{m \times k}$ with: $V$ ...
5 votes
1 answer
275 views

Are stably equivalent quadratic forms over Z equivalent?

Let $Q_1, Q_2, R$ be quadratic froms over $\mathbb{Z}$ such that $Q_1 \oplus R \cong Q_2 \oplus R$ as quadratic forms. Is it necessary that $Q_1 \cong Q_2$? I know that by Witt's theorem it is true ...
1 vote
0 answers
286 views

Generalized eigenvalue problem with nonnegative eigenvector constraint

Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution): $\underset{w}{\text{maximize}}\quad w^{\top}...