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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5
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4answers
503 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 ...
3
votes
2answers
360 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
100 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
71 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
159 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
38 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 ...
1
vote
7answers
387 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
100 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
115 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
133 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
56 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
76 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
60 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
374 views

What is the $\lVert$ symbol?

I am trying to understand the quadratic equation below but cannot understand what the double bars stand for. $$\lVert W_L LP' \rVert^2 + \sum_i W_{H,i}^2 \lVert p_i' - p_i\rVert$$
3
votes
1answer
843 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 ...
2
votes
1answer
1k views

Ellipse in Quadratic Form: Finding Intercepts with Principal Axes

Where an ellipse is expressed in quadratic form (e.g. $ax^2 + bxy + cy^2 = k$ is expressed as $x^TQx = k$), the principal axes are in the directions given by the eigenvectors of Q. I understand this. ...
1
vote
3answers
837 views

Positive-Definiteness of a Quadratic Form Matrix

I'm having trouble with some maths regarding the expression of the matrix quadratic form (i.e. $x^TAx$) and the proof that, where the eigenvalues $\lambda_1,\lambda_2,...,\lambda_n$ are all positive, ...
1
vote
2answers
541 views

Computing the rank and signature of a quadratic form - quick way?

Is there a 'quick way' of computing the rank and signature of the quadratic form $$q(x,y,z) = xy - xz$$ as I can only think of doing the huge computation where you find a basis such that the matrix of ...
1
vote
2answers
230 views

Isometry without injection and surjection

Suppose that $B_1$ and $B_2$ are bilinear form on space $V_1$ and $V_2$. An isometry relative to $B_1$ and $B_2$ is an linear map $\sigma:V_1 \rightarrow V_2$ satisfying ...
0
votes
1answer
103 views

example of quadratic form with nonunique +ve and -ve definite subspaces

What is an example of a quadratic form $Q$ on real vector space $V$ such that the maximal positive definite subspace and the maximal negative definite subspace are not uniquely determined? By ...
0
votes
3answers
174 views

Numbers of the form $x^2+axy+by^2$

This book, which needs to be returned quite soon, has a problem I don't know where to start. How do I find a 4 parameter solution to the equation $x^2+axy+by^2=u^2+auv+bv^2$ The title of the ...
4
votes
2answers
300 views

Equivalence of quadratic forms over p-adic fields.

There is a theorem that states that two quadratic forms over $\mathbb{Q}_p$ are equivalent iff they have the same rank, discriminant and the same $\epsilon$ invariant. (The last is defined as ...
4
votes
1answer
273 views

Map preserving indefinite scalar product must be linear

Let $V$ be a finite dimensional real vector space and $\langle\cdot,\cdot \rangle$ be a positive definite scalar product in $V$. It is well know that if a map $T:V \to V$ preserves ...
6
votes
0answers
96 views

Proving the multiplicativity of a quaternary quadratic form

Consider the set $S$ of all integers of the form $f(x,y) + f(z,w)$, where $x,y,z,w$ are integers, $$ f(x,y) = a x^2 + b x y + a^2 y^2,$$ and $a,b$ are integers with $$a > 1, \; \; 0 < b < ...
2
votes
1answer
344 views

Sesquilinear form/inner product

I have just a quick question: what is the relationship/difference between a sesquilinear form $F(u,v)$ and a complex inner product $(u,v)$?
3
votes
1answer
6k views

Quadratic equation -> matrix?

Problem: Find the EigenValues and EigenVectors of the matrix associated with quadratic forms $2x^2+6y^2+2z^2+8xz$. I know how to convert a set of polynomial equations to a matrix but I have no clue ...
4
votes
2answers
104 views

Proving the multiplicativity of a binary quadratic form

Consider the set $S$ of all integers of the form $x^2+y^2+4xy$, where $x$ and $y$ are integers. How could one prove the set $S$ is closed under multiplication? I have tried the bashy brute force ...
5
votes
1answer
151 views

Is there a nice way to define the “maximum” of two quadratic forms?

Suppose I have two quadratic forms on $\mathbb R^n$, represented as symmetric matrices $A$ and $B$ on the usual basis. I am interested in approximating the function $x \mapsto \max(x^TAx, x^TBx)$ ...
8
votes
1answer
144 views

On the set of integer solutions of $x^2+y^2-z^2=-1$.

Let $$ \mathcal R=\{x=(x_1,x_2,x_3)\in\mathbb Z^3:x_1^2+x_2^2-x_3^2=-1\}. $$ The group $\Gamma= M_3(\mathbb Z)\cap O(2,1)$ acts on $\mathcal R$ by left multiplication. It's known that there is ...
1
vote
3answers
2k views

Determine if the quadratic form is positive definite, negative definite or undefinite

"Determine if the following quadratic form is positive definite, negative definite or undefinite $Q:R^3\to R, Q(u)=x_1^2+4x_1x_2-2x_2^2+2x_1x_3-2x_3^2$" $$Q=\begin{bmatrix} 1&2&1 \\\ ...
1
vote
1answer
134 views

Need reference about quadratic forms on abelian groups.

Let $B$ and $C$ be abelian groups (in additive notation). We call a function $f:B\rightarrow C$ a quadratic form if for all $x,y,z \in B$, the function $f$ satisfies the relation ...
2
votes
1answer
155 views

A question regarding local minimizer of a function restricted on a circle

I have a quadratic function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f(\mathbf{x}) = (\mathbf{x}-\mathbf{p})^\top \mathbf{Q} (\mathbf{x} - \mathbf{p})$ where $\mathbf{Q}$ is positive definite and ...
0
votes
1answer
482 views

Canonical form of a Matrix

My understanding of canonical form is very limited, and so may require some help. Suppose a quadratic of the form: $$ x_1*x_2+x_1*x_3=Q.$$ How would one go about putting that into canonical form, ...
0
votes
1answer
1k views

Proving that a matrix is negative definite using its principal minors

I am interested to find out the proof for the following statement (it's from my textbook and it is stated without proof): A symmetric matrix is negative definite if and only if all of its ...
4
votes
1answer
479 views

When are two diagonal matrices congruent?

This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
1
vote
1answer
395 views

Quadratic forms and prime numbers in the sieve of Atkin

I'm studying the theorems used in the paper which explains how the sieve of Atkin works, but I cannot understand a point. For example, in the paper linked above, theorem 6.2 on page 1028 says that if ...
5
votes
3answers
388 views

Etymology of the word “isotropic”

Given a quadratic form $q : V \rightarrow k$, a nonzero vector $v \in V$ is said to be isotropic if $q(v) = 0$. Any subspace of $V$ containing such a vector is also said to be isotropic, and the ...
1
vote
1answer
116 views

Quadratic form over the dyadic numbers

I would like to know whether $q=\langle 3,3,11\rangle$ (a diagonal ternary form) represents $2$ over $\mathbb{Q}_2$ (i.e. whether there exist $x,y,z\in\mathbb{Q}_2^\times$ such that $q(x,y,z)=2$). I ...
3
votes
2answers
341 views

Isotropy over $p$-adic numbers

Over what $p$-adic fields $\mathbb{Q}_p$ is the form $\langle3, 7, -15\rangle$ isotropic?
3
votes
2answers
238 views

Hilbert Symbols when $K =$ the $p$-adic numbers

How can I show that the Hilbert Symbol is bimultiplicative, when the local field is the $p$-adic numbers? Everything I can find just sort of asserts bimultiplicativity without much proof, so I'm ...
1
vote
2answers
100 views

Showing: If $w\in C\ell^1(V,Q)$ anticommutes with all $v\in V$, then $w=0$

Show that if an element of the odd part of the Clifford Algebra anticommutes with everything in the vector space, then it is 0. Been having a really hard time making any progress with this one.
1
vote
1answer
153 views

Find an optimal 4-tuple which satisfies a boolean expression

(I have post a question with bounty for several days (Find a best 4-tuple which fulfils a variable boolean formula), but unfortunately got no answer yet. Here I simplify it to a smaller problem and ...
12
votes
1answer
481 views

Polarization: etymology question

The polarization identity expresses a symmetric bilinear form on a vector space in terms of its associated quadratic form: $$ \langle v,w\rangle = \frac{1}{2}(Q(v+w) - Q(v) - Q(w)), $$ where $Q(v) ...
1
vote
1answer
325 views

Find all solutions of this diophantine equation of the second degree in three variables

Consider the Diophantine equation $Q(x,y,z)=1$, where $Q(x,y,z)$ is the quadratic form $x^2+y^2-z^2$. Let $S \subseteq {\mathbb Z}^3$ denote the set of all solutions. It is rather easy to find several ...
2
votes
1answer
73 views

Another Witt ring question

Let $F$ be a field of characteristic not equal to to $2$, $W(F)$ Witt ring of the quadratic forms. I've been trying to prove that $I^2(F)=0$ implies that every binary quadratic form over $F$ is ...
9
votes
4answers
397 views

Representing a number as a sum of at most $k$ squares

Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to $$ n = x_1^2 + ...
2
votes
0answers
122 views

Quadratic transformations of vector spaces

Much is known about transformations of the following form $$y_i = L_{ij}x_j \;\;: \;\; x\in\mathcal{R}^n, L\in\mathcal{R}^{n\times n}$$ We can infer a number of geometric properties about the ...
0
votes
1answer
105 views

Quadratic spaces and definition of hyperbolic plane

I'm trying to figure out a proof in Lam's book on quadratic forms (he uses this to define a hyperbolic plane). He states that if $(V,q)$ is a 2-dimensional quadratic space over $F$, then the following ...
9
votes
2answers
568 views

Proving an inequality with Cauchy-Schwarz

In the "User's guide to viscosity solutions" by Crandall, Ishii and Lions (link), they make the following claim (inequality (A.4) p. 58) : Given $x$, $\xi$ $\in \mathbb{R}^n$, $A \in \cal{S}(n)$ ...
1
vote
1answer
103 views

Extension to Witt's bases

Let $X$ be a real finite dimensional linear space. Let $B:X\times X \rightarrow \mathbf{R}$ be a bilinear symmetric non-degenerated form. Let $M$, $N$ be totally isotropic subspaces with the same ...