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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1
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3answers
194 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
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1answer
120 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
133 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
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1answer
232 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
41 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: ...
3
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0answers
541 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
94 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
429 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 ...
1
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1answer
60 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
152 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 ...
4
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1answer
147 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 ...
2
votes
2answers
3k 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
159 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
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3answers
151 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 ...
6
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1answer
161 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
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0answers
65 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 ...
4
votes
1answer
230 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 ...
1
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2answers
355 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
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4answers
415 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
309 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
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1answer
96 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
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1answer
142 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 ...
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1answer
35 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
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7answers
357 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
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4answers
99 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
126 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
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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
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1answer
72 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
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1answer
59 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
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1answer
359 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
746 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
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3answers
743 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
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2answers
487 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
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2answers
224 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
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1answer
97 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
171 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
276 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
246 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
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0answers
92 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
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1answer
280 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 $(u,v)$?
2
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1answer
5k 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
98 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
140 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
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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
130 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
152 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 ...