Questions about quadratic forms in many variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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2
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4answers
30 views

Is the matrix $A$ positive (negative) (semi-) definite?

Given, $$A = \begin{bmatrix} 2 &-1 & -1\\ -1&2 & -1\\ -1& -1& 2 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Define the ...
-1
votes
1answer
40 views

How many ways can a quadratic form represent a prime?

Given $a,b,c,p\in\Bbb N$ with $b^2-4ac<0$ and $p$ is a prime with $\bigg(\frac{b^2-4ac}p\bigg)=1$, how many solutions $(x,y)\in\Bbb Z^2$ are there to $$ax^2+bxy+cy^2=p?$$
0
votes
1answer
23 views

How to transform the quadratic form of an ellipse to a circle

Consider the ellipse $x^TPx\le a$. I would like to transform (the quadratic form of) this ellipse into a circle $y^T\begin{pmatrix}1&0\\0&1\end{pmatrix}y\le b$ via a coordinate transform ...
1
vote
1answer
23 views

quadratic form in hilbert space and Gram matrix

We are in Hilbert space $L^2$ we are given a subspace of dimension K as $$ V=\{ g_k,1 \le k \le K \} $$ everything that folows is defined on $V$ we define map $$ x \mapsto Q(x):= \sum_{k=1}^{K} ...
0
votes
1answer
30 views

Quadratic form in Hilbert space associated with orthogonal projection operator

we are in Hilbert space $L^2 $ and we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
0
votes
0answers
8 views

Moments of quadratic forms

$x=(x_1,...,x_T)'$ is a $T\times1$ random vector, where $x_t, t=1,..., T$, is a stationary process with mean zero and finite fourth moments. $A$ is a $T\times T$ symmetric constant matrix. How to find ...
0
votes
4answers
59 views

Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.

I try to transform Transform $$f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$$ to a diagonal form. I can do it using eigenvalue, but when I directly complete the square to find its ...
3
votes
2answers
86 views

Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
0
votes
0answers
32 views

Simultaneously diagonalise two real quadratic forms

I would like to simultaneously diagonalise the quadratic forms $A=2x^2+3y^2+3z^2-2yz$, and $B=x^2+3y^2+3z^2+6xy+2yz-6zx$. Of course there's a theorem saying this is possible and I followed the ...
2
votes
1answer
38 views

How to compute the matrix $S$ in Sylvester's law of inertia

Sylvester's law of inertia states that for any symmetric matrix $A$ there exist an invertible matrix S such that, $S^T A S = D$, where $D$ is a diagonal matrix which has only entries 0, +1 and −1 ...
0
votes
2answers
21 views

Can I find a vector with entirely nonzero entries for the following quadratic form to evaluate to zero?

I have a square, symmetric matrix $M$, of size at least $2\times 2$, with diagonal entries equal to $1$ and off-diagonal entries equal to $\pm 1$. Let the entries of $M$ be such that it is indefinite. ...
2
votes
1answer
23 views

Omission in Jacobson's BAI regarding extension of isometries.

Suppose $V$ is a finite dimensional vector space over a field of characteristic $\neq 2$ equipped with a nondegenerate quadratic form $Q$. Witt's cancellation theorem says that if $U_1,U_2$ are ...
3
votes
2answers
91 views

Multivariate Gaussian integral of ratio of quadratic forms

Given two real symmetric matrices $M,S$ is there a known answer for the Gaussian integral $\int d^Nz\frac{z^TMz}{z^TSz}$ where the integration is over N-dimensional Gaussian variable $z\sim ...
5
votes
3answers
116 views

reference for linear algebra books that teach reverse Hermite method for symmetric matrices

The method I mean is useful for symmetric matrices with integer, or at least rational entries. It diagonalizes but does NOT orthogonally diagonalize. The direction I do it, I usually call it Hermite ...
0
votes
0answers
19 views

Minimizing a quadratic form with orthogonality constraints

Suppose $A$ is an $n$-by-$n$ symmetric matrix, and I want to find $x_{1}$ and $x_{2}$ that maximize $x_{1}^{T} A x_{1} + x_{2}^{T} A x_{2}$ subject to the constraint that $x_{i}^{T} x_{j} = ...
0
votes
0answers
18 views

Find Isotropic vectors that form a basis

I have this question: let $(E,\langle,\rangle)$ an inner product space with dimension $n$ and $u$ a symmetric linear transformation and we define a quadratic form $q$ by $$\forall x\in E,\quad ...
0
votes
2answers
28 views

Solve the algebraic expression for a, b, and c of the function x

I am trying to solve for $a$, $b$, and $c$ in the expression below, but I have found that the way I tried to solve it is convoluted and did not work out. I believed that by solving for x, I would be ...
4
votes
1answer
119 views

Reducing a pair of indefinite quadratic forms to the canonical form

Assume $A, B$ being a pair of symmetric matrices over reals. Let $$ \varphi_1(x) = (x, Ax)\\ \varphi_2(x) = (x, Bx). $$ There's a well-known result that if $A > 0$ then the pair of forms can be ...
5
votes
3answers
106 views

Parametrization of $x^2+ay^2=z^k$, where $\gcd(x,y,z)=1$

$x,y,z$ be three coprime integers, $a \in \mathbb{Z}>0$ and $k$ an odd integer. How do I find all the non-trivial solutions of the diophantine equation? $$x^2+ay^2=z^k$$ Does the method which ...
5
votes
0answers
51 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
5
votes
2answers
98 views

How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...
1
vote
1answer
31 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
1
vote
1answer
37 views

Quadratic form as generalized distance?

In the book A Linear Systems Primer (by Antsaklis and others), they first mention squared distance of a point x from the origin: $$x^{T}x = ||{x}||^2$$ which represents the square of the ...
0
votes
2answers
46 views

Can this expression be made into a quadratic form?

Can this expression be made into a quadratic form: $ a x_t -\gamma {x_t}^2 $ I want to solve a linear quadratic programming problem and it requires that I put this expression in a quadratic form. $ ...
0
votes
0answers
12 views

How to do linear quadratic dynamic programming with non homogeneous quadratic equation

I am not well versed on matrix algebra and linear quadratic programming. I am wondering if it is possible to make a non-homogeneous equation into a homogeneous one. I need to make the following ...
0
votes
1answer
17 views

Almost universal integer quadratic forms

This question is inspired by the 15-theorem. For any nonnegative integer k, define a k-universal integer quadratic form to be a form that represents all but k positive integers. So, universal forms ...
0
votes
0answers
42 views

Question about Mumford's article

I'm reading the following article by Mumford speaking about theta characteristic. Mumford's article I'm trying to understand the definition af the quadric form $q$ on page 184. Here my questions: 1) ...
0
votes
0answers
28 views

how to determine a biquadratic form is positive-definite

A biquadratic form $\sum_{i,j,k,l}b_{i,j,k,l}x_{i}x_{j}y_{k}y_{l}$, how to determine whether it is positive-definite? A necessary and sufficient condition? In fact, I have a matrix $B=\sum_{1\leq ...
3
votes
0answers
28 views

Quadratic form and matrix

We know quadratic form $f(x_1,x_2)= a_{11} x_1^2 + 2 a_{12} x_1 x_2 + a_{22} x_2^2$ is non-negative for all $x_1,x_2 \in \mathbb{R}$ iff matrix $(a_{ij})_{2 \times 2}$ is semi-positive defined. My ...
0
votes
0answers
27 views

Complexity of computing a posiform of a quadratic pseudo-boolean function

I am reading the chapter 13, Pseudo-Boolean functions, of Boolean Functions: Theory, Algorithms, and Applications by Crama et. al. In section 13.2, the authors introduce the idea of Posiform. The ...
-1
votes
2answers
20 views

graph quadratic form and find the equation of asymptotes

So I had this quadratic form that need to be graphed showing both original and new axes. And I also need to find out the equation of asymptotes. $$ \left\{ \begin{aligned} ...
0
votes
1answer
20 views

Maximization of quadratic form over complex unit cube

I am trying to find the maximum of a hermitian positive definite quadratic form $xQx^H$ (where $Q=Q^H$ and all eigenvalues of $Q$ are non-negative) over the complex unit cube $|x_i|\leq 1$, ...
1
vote
0answers
28 views

Differentiating quadratic form containing vector raised to powers elementwise, can I avoid Hadamard notation?

Say $\mathbf{M}$ is a symmetric, p.d. 2x2 matrix, and $\mathbf{x}$ is a 2x1 vector. The familiar quadratic form is of course given by: $A=\mathbf{x'}\mathbf{M}\mathbf{x}$ (where $A$ is a scalar), and ...
0
votes
1answer
37 views

existance of a solution to quadratic form equation

Let $\lambda$ is an unknown scalar and; $Q=Q_1 - \lambda*Q_2$ where $Q_1, Q_2$ are $NxN$ positive symetric matrices, $B=B_1 - \lambda*B_2$, where $B_1, B_2$ are $Nx1$ vectors, $m=m_1 - ...
0
votes
2answers
38 views

Minimum quadratic form value within a line?

If I have $x\in R^n , C\in R^{m\times n}, d\in R^m$, $m<n$, then $Cx=d$ is a linear manifold. And $P\in R^{n\times n}$, $P>0$, the quadratic form is $y=x^TPx$ Is there an analytical expression ...
0
votes
0answers
30 views

Semidefinite relaxation of Quadratic equation

I have read in various papers that we can write a Quadratic equation with symmetric matrix as a linear programming problem. For example $$f(x)= x^T*Q*x + c$$ where Q=[2 0;0 3]; Now we can write ...
1
vote
0answers
15 views

Reduction of positive definite binary quadratic forms over congruence subgroups

Let $\Gamma_0(N)$ be a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and $Q(x,y)$ be a positive definite binary quadratic form with leading coefficient $a$ divisible by $N$. Can someone give me a ...
2
votes
2answers
82 views

Regular Quadratic Space - isotrope vector

I am currently trying to solve the following exercise: Show that every regular quadratic space of finite dimension $E$ that contains at least one isotrope vector, has a basis consisting only of ...
0
votes
0answers
10 views

non-uniform arithmetic lattice in a semisimple algebraic group over a local field of positive characteristic

Say I'm considering the group of rational points $G(k)$ where $G$ is the special orthogonal group for the quadratic form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}-x_{5}^{2}$ and $k$ is a ...
0
votes
0answers
17 views

example of a quadratic form

Would someone be able to tell me an example of a quadratic form defined on a five-dimensional vector space $V$ over a non-archimedean local field $k$ of positive characteristic (not equal to two, say) ...
2
votes
0answers
20 views

What is binary norm of quadratic fields of sum of two squares such that one of them is necessarily even like $a^2 +4b^2?$

I am trying to simplify an expression which I have reached, suppose a number can be represented in the form of $D=a^2 + 4b^2$. What is binary norm of $D$, or how else can it be represented?
0
votes
0answers
9 views

meaning of index of a quadratic form for a field of positive characteristic

In "Classification of Algebraic Semisimple Groups" in "Algebraic Groups and Discontinous Subgroups: Procedings of Symposia in Pure Mathematics, Volume IX", Jacques Tits speaks of the index of a ...
0
votes
0answers
8 views

Perturbed quadratic form definiteness

I have a rectangular matrix $[\mathbf \Pi]\in \mathbb R^{m\times n}$, where $m>n$, and $\mathrm {rank}([\mathbf \Pi])=k<n$. Its left null space basis are the columns of $[\mathbf Z]$. It's ...
1
vote
1answer
33 views

Tricky problem about quadratic forms

Let A be symmetric matrix of order 3. Consider set $S=\{x\in\mathbb{R}^{3}:\; x^{T}Ax=a\}$. 1 (true). If S is unbounded for any $a\in\mathbb{R}$, then A is indefinite. 2 (false). If S ...
0
votes
0answers
33 views

Hermitian form of a unitary space

If $V$ is a finite $n$-dimensional complex vector space with a hermitian form $h$ then $h$ is given by a hermitian matrix $A$ with the transformation law $P^{t}A\bar{P}$ where $P$ is an invertible ...
0
votes
1answer
18 views

Diagonalizing quadratic forms

I know how to diagonalize a given quadratic form using the Gaussian method. Though, I once read somewhere that there's another method which uses an augmented matrix, but I didn't go into details. I ...
2
votes
2answers
32 views

find real numbers so that a specific linear isometry exists

Consider the quadratic form $$f: \mathbb{R}^3 \to \mathbb{R}, (x_1, x_2, x_3) \mapsto 3x_1^2 - 3x_2^2 + x_3^2-2x_1x_3$$ I want to find $\lambda_1, \lambda_2, \lambda_3 \in \mathbb{R}$, so that there ...
8
votes
1answer
85 views

Why does the plot of the legendre symbol of $x^2 - y^2$ over a finite field look rectangular

The small top-left thing is a plot of the legendre symbol of $x^2 - y^2$ over $\Bbb F_{37}$. The thing in the middle is plot for $\Bbb F_{587}$. The thing on the right is a plot of the legendre ...
0
votes
1answer
14 views

Show that $|Q(a,b)| \leqslant M \cdot \max(|a|^2,|b|^2).$

Prove if $Q(x,y)=Ax^2+Bxy+Cy^2$ is a quadratic form, then the constant $M = |A|+|B|+|C|$ satisfies the property that $|Q(a,b)| \leqslant M \cdot \max(|a|^2,|b|^2)$ for every $a,b \in \mathbb{R}.$ ...
0
votes
0answers
17 views

Quadratic Form for Matrix with Sum

I just had a pretty basic question. Suppose you have the quadratic form: $$(u+x)^t V (u+x)$$ If you expand this out, you get something along the lines of $$u^t V u + x^t V x + u^t V x + x^t V u$$ ...