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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132 views

Quadratic Forms and Congruences

How does one prove (the non-trivial direction) that, for $n \in \mathbb{N}$, $x^2 + y^2 + z^2 = n$ solvable $\iff$ $x^2 + y^2 + z^2 \equiv n\ (m)$ solvable for all $m$? In particular, is there a ...
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2answers
57 views

Showing representation numbers are at most on the order of polynomial growth

If $Q$ is the sum of squares quadratic form $\sum_1^n x_i^2$ over some lattice, then $r_Q(m)$, the number of representations of an integer $m$ by $Q$ (order/sign matter) is sometimes given in a nice ...
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1answer
896 views

Why do we assume that a matrix in quadratic form is Symmetric?

I am looking to the review document for linear algebra and the part of the quadratic form (pg17) mentions about an assumption of being symmetric for a matrix in quadratic form. It also includes some ...
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0answers
114 views

counterexample for: integral forms that are equivalent as rational forms are also equivalent as integral forms

I have the feeling that I'm missing something very obvious: I'm looking for a counterexmple for the following statement for some $n>1$ (it is trivially true for $n=1$): Let $A,B\in\mathbb ...
4
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0answers
317 views

Understanding a proof about Hilbert Matrix

EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO. Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ...
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0answers
251 views

quadratic form of trace_inverse of symmetric positive definite matrix

I have the following problem: I need to implement a program that doesn't accept the matrix quadratic form $B^T\times B$ but it accepts the scalar quadratic form instead. Actually I need to find a ...
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2answers
365 views

A standard quadratic minimization problem

Consider the "Complex" Quadratic minimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1 \end{align} ...
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4answers
330 views

Find $a$ and $b$ in the given cubic polynomial

Find $a$ and $b$ such that $x+1$ and $x+2$ are factors of the polynomials $x^3+ax^2-bx+10$. Here I am not sure that how can I obtain the value of $a$ and $b$, I tried to multiply $x+1$ and $x+2$ to ...
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4answers
313 views

Derive the Quadratic Equation

Find the Quadratic Equation whose roots are $2+\sqrt3$ and $2-\sqrt3$. Some basics: The general form of a Quadratic Equation is $ax^2+bx+c=0$ In Quadratic Equation, $ax^2+bx+c=0$, if ...
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1answer
92 views

Need for algorithm on solving a set of quadratic matrix?

Firstly, I want to thank @adam W gives a good clue to solve my homework problem. I have a set of quadratic matrix need to solve(not one equation) according to the following form: ...
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2answers
516 views

Change parabolic equation to canonical form

I have equation $y = -x^2 + 2x + 7$. How can I change it to canonical form, which looks like $y^2 = 2px$ ? ($p$ will be parameter) What i ve tried so far: $$\begin{align} y &= -x^2 + 2x + 7\\ y ...
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4answers
191 views

Some weird equations

In our theoreticall class professor stated that from this equation $(C = constant)$ $$ x^2 + 4Cx - 2Cy = 0 $$ we can first get: $$ x = \frac{-4C + \sqrt{16 C^2 - 4(-2Cy)}}{2} $$ and than this one: ...
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2answers
1k views

Show $15x^{2} - 7y^{2} = 9$ has no integer solutions

I'm trying to show the quadratic binary has no integer solution. I've used the following process to transform it into a Pell's equation of the form $x^{2} - Dy^{2} = M$ If there is a solution, then ...
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0answers
109 views

Minimize a complex quadratic subject to two convex quadratic constraints

I have the following the optimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\} \\\ ...
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1answer
399 views

A good reference for Quadratic Forms

Can anyone recommend a good reference for brushing up on quadratic forms? They keep coming up (quite naturally of course) in the context of differential geometry and I find I am rustier than I ...
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0answers
95 views

quadric surfaces. Uniqueness theorem

I would like to know references (book or monograph) for this theorem about quadric surfaces. If $F(x)=0$ is an equation in $K[x]$ of a quadric surface $C$ and i) ${\rm char}(K) \not= 2$ ii) $C$ has a ...
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5answers
791 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 ...
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2answers
96 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. ...
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1answer
246 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. ...
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2answers
594 views

How to deal with translations, with change of coordinates for Quadratic Forms: $x^TAx=k$ ?

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 ...
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182 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 ...
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2answers
778 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 $$ ...
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36 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 ...
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1answer
90 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 ...
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0answers
138 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 ...
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1answer
156 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 ...
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3answers
2k views

Hessian matrix of a quadratic form

I need a help with one example. I have to proove that hessian matrix of a quadratic form $f(x)=x^TAx$ is $f^{\prime\prime}(x) = A + A^T$. I am not even sure how the Jaxobian looks like (I never did ...
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1answer
255 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 $x$ ...
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1answer
135 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 ...
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1answer
73 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 ...
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3answers
193 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 ...
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3answers
215 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. ...
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1answer
135 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 ...
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1answer
154 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?
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1answer
234 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: $${a}^{2}+{b}^{2}$$ ...
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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: ...
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671 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 ...
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2answers
96 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 ...
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1answer
529 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 ...
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1answer
67 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 ...
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0answers
162 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 ...
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1answer
151 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 ...
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2answers
4k 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 ($\Delta_X X^TAX$) ...
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164 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. ...
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3answers
156 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 ...
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1answer
171 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 ...
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0answers
66 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 ...
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1answer
242 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 ...
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2answers
366 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
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 ...