Tagged Questions

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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10
votes
5answers
315 views

Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$

Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
2
votes
2answers
57 views

Existence of complex solutions satisfying two quadratic forms

If I have two linear equations, $ax + by = 0$ and $cx + dy = 0$, and I wanted to find out if they had any non-trivial solutions, I would simply check if $(a,b)$ and $(c,d)$ are linearly dependent. ...
0
votes
1answer
67 views

How do you construct a lattice from its basis or its Gram Matrix?

I'm really having trouble trying to understand this. A few weeks back, I got pretty interested in sphere packing and I'm trying to grasp the idea of using a matrix to represent the basis of a lattice. ...
1
vote
2answers
208 views

Linear Algebra Application to Quadratic Forms

When working with a change of coordinates using $x^TAx=k$ how and when do we deal with translations? I'm comfortable with setting up the formula $x^TAx$ where A is the matrix whose diagonal ...
5
votes
0answers
87 views

What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS In December 2010 my question appeared in the M.A.A. Monthly, show that $4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 ...
1
vote
2answers
163 views

non-symmetric positive definite matrix!?

Is symmetry a necessary condition for positive (or negative) definiteness? If not: It can be proved that if $\mathbf{A}:(m\times m)$ is a square (non-symmetric) matrix, then $$ ...
0
votes
0answers
32 views

two non-degenerate quadratic forms on $GF(2)^2r$

I know this: There are two non-degenerate quadratic forms on $GF(2)^2r$. The hyperbolic form may be taken to be $Q^+(x)=x_0 x_1 + \cdots +x_{2r-2}x_{2r-1}$ , and the elliptic form to be ...
1
vote
1answer
56 views

Solution count of quadratic form congruence over $\Bbb Z / 8 \Bbb Z$

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
1
vote
0answers
69 views

solution count of quadratic form congruences

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
4
votes
1answer
129 views

Centre of a quadric

I found the following sentence in my linear linear algebra book (affine and projective geometry): $Q:V \to \mathbb{K}$ is a quadric (quadratic function) and $\alpha\in Aff(V)$. $Aff(V)$ is the set of ...
4
votes
1answer
90 views

Matrix Equation with Quadratic form

I am working in a problem that involves multivariate normal distributions and, at a given point, I need to solve the following matrix equation: $$x=\sqrt{x^{\prime}\Sigma^{-1}x} \cdot y$$ Where $x$ ...
6
votes
1answer
81 views

Completing squares by symplectic transformations

A quadratic polynomial of $2n$ variables is given as $$ H = \sum_{i,j=1}^{2n} A_{ij} x_i x_j = x^T A x, $$ where $A$ is a symmetric matrix. I am looking for a symplectic transformation of these ...
1
vote
1answer
55 views

Reference request for quadratic form diagonalization

I want to read a proof of "Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form" using Gram-Schmidt orthogonalization. Could anyone ...
4
votes
3answers
118 views

The quadratic form $x^2 + ny^2$ via prime factors

Elementary algebra shows that the product of two numbers in the form $x^2 + ny^2$ again has the same form, since if $p = (a^2 + nb^2)$ and $q = (c^2 + nd^2)$, $$pq = (a^2 + nb^2)(c^2 + nd^2) = (ac ...
1
vote
3answers
132 views

Matrix of a quadratic form?

What exactly is the matrix of a quadratic form? I have seen this notation occuring in a few papers (e.g. Siegel's unreadable German papers), with particular reference to the trace of a quadratic form. ...
1
vote
1answer
61 views

Independence of quadratic forms

Let us consider the quadratic form $$q_1 = \mathbf{x}_1^\mathrm{H} \mathbf{A}\,\mathbf{x}_1 $$ and the quadratic form $$ q_2= \mathbf{x}_2^\mathrm{H} \mathbf{A}\, \mathbf{x}_2 $$ where ...
1
vote
1answer
62 views

Transformation of Quadric Surfaces

Is there a transformation $T: \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ such that a hyperboloid of one-sheet can be mapped to a hyperboloid of two-sheets using such transformation?
2
votes
1answer
182 views

Geometric Significance of the Addition of Square Roots of Two Numbers

In a calculation, I've come across a relation along the lines of this: $${a}^{1/2}+{b}^{1/2}$$ My presumption would be that this is somewhat related to the Pythagorean relation: $${a}^{2}+{b}^{2}$$ ...
0
votes
2answers
39 views

Convexity of a function and constraint

Consider the quadratic function $f(x_1,x_2,x_3,x_4)=x_1+2x_2+4x_4+x_1^2+5x_2^2+3x_3^2x_4^2-4x_1x_2-2x_2x_3+2x_3x_4$. Is f a convex function? Consider a constraint defined using the above function f: ...
2
votes
0answers
202 views

Simultaneous diagonalization of quadratic forms

I would like to collect references (or direct quotations) about as many "simultaneous diagonalization" results in linear algebra as possible. Let $V$ be an $n$-dimentional ($n$ finite) vector space ...
3
votes
2answers
86 views

When does a binary quadratic form represent 1 or -1

Let $a,b,c$ be integers. Is there a reasonably concise condition on $(a,b,c)$ which ensures that $$ax^2+bxy+cy^2=\pm 1$$ has a solution in integers $x,y$? In addition to direct answers I would also ...
3
votes
1answer
120 views

Why does positive semi-definiteness in this inequality imply a convex set?

I was reading a proof that rewrote an inequality in the form: $$b^Tx +x^T A x \le \alpha$$ for $b,x \in \mathbb{R}^n$ and $\alpha \in \mathbb{R}$, and with $A$ positive semidefinite. It then ...
0
votes
0answers
19 views

Solving for bayesian decision surfaces-quadric vector equation

I have three discriminant functions g1(x) , g2(x) and g3(x), each denoting one of the three classes under which an input is to be specified. The class conditional densities are gaussian. The ...
1
vote
1answer
48 views

Quadraticize a generic function

I have a generic function: $g(s,u)$. Now I want to have a local approximation near the point $(s^{\star}, u^{\star})$ in the quadratic form $$s^{T} Q s + u^{T} R u$$ to apply an optimal control ...
2
votes
0answers
100 views

A characterization of an ambiguous class of binary quadratic forms of discriminant $D$

We use the definitions of this question. Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). There exists a bijection $\psi\colon Cl^+(R) \rightarrow C(D)$ by ...
0
votes
0answers
46 views

The inverse class of the class represented by a primitive binary quadratic form of discriminant $D$

We use the definitions of this question. Is the following proposition true? If yes, how do we prove it? Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ ...
4
votes
1answer
127 views

Help fixing my broken example of Arf invariant

I need help fixing a broken example I've come up with. In particular, I wanted to use the Arf invariant to distinguish two non-homeomorphic surfaces. That's the first part that's broken since there ...
1
vote
1answer
642 views

Derivative of Quadratic Form

For the Quadratic Form $X^TAX; X\in\mathbb{R}^n, A\in\mathbb{R}^{n \times n}$ (which simplifies to $\Sigma_{i=0}^n\Sigma_{j=0}^nA_{ij}x_ix_j$), I tried to take the derivative wrt. X ($\Delta_X X^TAX$) ...
4
votes
0answers
120 views

Proof that the Arf invariant is independent of choice of basis

I'm confused about the proof of the following claim: Let $V$ be a vector space of dimension $2n$ and let $e_i, f_i$ be a symplectic basis. Let $q: V \to Z_2$ be a non-degenerate quadratic form. ...
3
votes
3answers
128 views

Question about answer about quadratic forms on MO

I have a question regarding this MO answer: The answer says that in characteristic $2$, we cannot obtain a quadratic form from a bilinear form. I thought it was the other way around and now I am ...
5
votes
1answer
117 views

Follow up on intersection forms

For which topological spaces $X$ can I define an intersection form $b(\cdot, \cdot)$? I know at least one example: If $X$ is a closed orientable $2n$-manifold then one can define an intersection ...
3
votes
0answers
60 views

Siegel's theorem

I want to learn the proof of the following theorem by Siegel. The statement of the theorem is taken from "Symmetric bilinear forms" by Milnor and Husemoller (pp. 44). They say that the proof is due to ...
3
votes
1answer
180 views

Question about $4$-manifolds and intersection forms

This is a question related to an earlier question of mine: I've been reading about topological invariants. Some of them are defined in terms of quadratic forms. My current understanding is: we can ...
0
votes
2answers
270 views

Finding positive integer solutions to $n = ax^2 +by^2 - cxy$

How can I find the positive integer solutions to $x$ and $y$, given that $n$, $a$, $b$ and $c$ are all positive integers, in an equation of the form: $$n = ax^2 + by^2 - cxy.$$ Specifically, I want ...
5
votes
4answers
232 views

Applications of quadratic forms

It seems that a lot of great mathematicians spent quite a while of their time studying quadratic forms over $\mathbb{Z},\mathbb{Q},\mathbb{Q_p}$ etc. and there is indeed a vast and detailed theory of ...
0
votes
0answers
110 views

Number of Totally Isotropic Subspaces

First I want to review some concept from quadratic form. Let $V$ be quadratic space over finite field $F$ and $char(F)\neq 2$ with quadratic form $q$. For exmaple $q:V\rightarrow V$ and $|F|=q$ and ...
2
votes
2answers
179 views

Solving a system of quadratic vector equations

This problem arises from my research in computer vision, specifically projective homography: I have $n$ unknown variables, represented by an $n\times 1$ vector $\mathbf{x}$. There is a system of $n$ ...
1
vote
1answer
85 views

Quadratic equation : Two solutions or one solution?

I have an equation to solve for y: $$\frac{y^2}{y}=1$$ Normally, I would cancel out one $y$ and get $y=1$ as a single solution. But If I think of it as quadratic equation $$y^2=y$$ $$y^2-y=0$$ ...
4
votes
1answer
67 views

Two elements $a$ and $b$ of a quadratic space generate a nondegenerate two dimensional subspace if and only if $(a\cdot a)(b\cdot b)\neq(a\cdot b)^2$

I whould like to prove the following statement: Lemma: Let $(V,Q)$ be a nondegenerate quadratic vectorspace over a field $\mathbb{F}$ and $a,b\in V\setminus\{0\}$. Then for ...
0
votes
1answer
96 views

General quadratic form of two variables

I was referring to this lecture http://www.stanford.edu/class/ee364a/videos/video04.html. and he gave an example of a generalized quadratic equation ...
0
votes
1answer
28 views

Optimization of Unconstrained Quadratic form

So I'm learning about optimization of quadratic forms and this textbook goes through definiteness of matrices and principle minors etc. and then goes straight onto optimizing with constraints but ...
0
votes
6answers
247 views

positive definite quadratic form

Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite? Approach: The matrix of this quadratic form can be derived to be the following $$M := \begin{pmatrix} 1 & ...
1
vote
4answers
91 views

Solution to a system of quadratics

I am learning about a Bell State, and am trying to show that they are entangled. I believe that the required proof is to show that the system $$\alpha_0^2+\alpha_1^2=1$$ $$\beta_0^2+\beta_1^2=1$$ ...
3
votes
1answer
106 views

Center of SO(V,q)

Let $V$ be finite dimensional vector spaces and $q$ is quadratic form. I'm looking for $Z(SO(V,q))$. where $SO(V,q)$ is special orthogonal group. If $\operatorname{dim} V$ is odd then ...
2
votes
1answer
86 views

Simultaneous Orthogonalization

Let $q,q':\mathbb V \longrightarrow \mathbb R$ be two quadratic form where $\mathbb V$ is vector space with $dim \mathbb V \geq3$ and $q(x)+q'(x)>0$ for any $0\neq x\in \mathbb V$ then there exists ...
1
vote
0answers
51 views

Witt Cancellation over $\mathbb{Z}/{p^e \mathbb{Z}}$?

I wonder whether someone knows if the Witt cancellation theorem also holds for the rings $\mathbb{Z}/{p^e \mathbb{Z}}$ where $p$ is an odd prime and $e \in \mathbb{N}$, i.e. for example, let $G = ...
0
votes
1answer
61 views

Action of $SL_2(Z)$ on Markoff quadratic forms

My setting is as follows: Fix a Markoff form $f_m(x,y)$ (see definition in the link below). If $f_m$ has the form ${\alpha}x^2+{\beta}xy+{\gamma}y^2$ then each element $A\in SL_2(Z)$ acts on $f_m$ in ...
1
vote
1answer
47 views

Quadratic Equation - What am I doing wrong?

For which value of $c$ does the quadratic equation $5x^2 - 6x + c = 0$ have exactly one solution in terms of $x$? The solution is supposed to be $c = 1.8$, but I only ever get $c = 1$ ...
1
vote
1answer
192 views

What is the $\lVert$ symbol?

I am trying to understand the quadratic equation below but cannot understand what the double bars stand for. $$\lVert W_L LP' \rVert^2 + \sum_i W_{H,i}^2 \lVert p_i' - p_i\rVert$$
2
votes
1answer
296 views

norm of a quadratic form

Suppose that $q$ is a quadratic form on $\mathbb{R}^n$, $q(x)=(x,Ax)$ say (or $q(x)=x^TAx$ if you prefer that notation). Then one could consider the quantity $$ \sup\{ \left|q(x)\right| : \left\| x ...

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