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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1answer
98 views

Quadratic Forms in Non-Linear Optimization

This is a rather trivial question but I am having a great deal of trouble: Let $f(x) = (1/2)xQx-xb$ and $E(x) = (1/2)(x-x^*)Q(x-x^*)$ then $E(x) = f(x) + (1/2)x^*Qx^*$ where $x,x^*,b$ are vectors ...
3
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1answer
151 views

In quadratic form, how would symmetric matrix $A$ would change under coordinate change?

In http://en.wikipedia.org/wiki/Quadratic_form, Let $q$ be a quadratic form defined on an n-dimensional real vector space. Let $A$ be the matrix of the quadratic form $q$ in a given basis. ...
2
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2answers
175 views

Finding a parametrization of a hyperbola who has a fixed signature

How do I find parametrization of the hyperbola $x^2-y^2=1$ which is the unit sphere of a quadratic form with signature $(1,-1)?$ The only parametrization that comes to mind is $x=\cosh t,y=\sinh t$. ...
1
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1answer
75 views

Determining a norm from a quadratic form

If $B$ is a quadratic form over some space $V$, what is the norm determined by $B$? Is this the inner product $\langle Bu,Bv\rangle$? If not, and it is not possible to determine a norm from knowing ...
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2answers
142 views

Lucky Lattice Points

How many lattice points lie on the sphere given by following equation ? $$x^2+y^2+z^2=2013$$ Hint: A lattice point has integer coordinates.
2
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0answers
34 views

$U$ is isotropic $\implies$ $U\subset U^0$

Let $(V,Q)$ be a quadratic module and $U$ is a subspace of $V$. Serre (A Course in Arithmetic, p. 29) claimed that the following is evident: $U$ is isotropic $\iff$ $U\subset U^0$. In other ...
3
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6answers
387 views

Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
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1answer
83 views

Rewriting a quadratic Matrix equation as a quadratic vector equation

Consider the set of $N \times N$ matrices $\{W_i\}_{i=1}^{i=L}$, set of $N \times 1$vectors $\{g_i\}_{i=1}^{i=L}$ and $\{h_i\}_{i=1}^{i=L}$. Now consider the following sum \begin{align} ...
1
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1answer
175 views

Find a matrix that simultaneously diagonalizes to matrices

struggling with a question from homework and would appreciate some assistance. Let $A, B \in M_2^{\mathbb{R}}$ be defined as follows: $$A = \begin{pmatrix} 2 & 1 \\ 1 & 1 ...
1
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1answer
163 views

Quadratic Forms

When defining a quadratic form why is it that we place $\frac{1}{2}$ in front? That is, why do we use $f(x) = \frac{1}{2}(x^T Qx) - b^T x$? Is this simply a convention that comes from the ...
1
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3answers
99 views

Sum of two quadratic forms

Suppose I have two quadratic forms $Q_i(x)=(x-a_i)^T A_i(x-a_i)+c_i$, $i=1,2$ where $x,a_i \in \Bbb{R}^n$ and $A_i$ are positive-definite $n\times n$ matrices. Then $Q(x)=Q_1(x)+Q_2(x)$ is also a ...
2
votes
1answer
116 views

Graphically, what is positive semidefinite-ness?

Suppose that we are trying to minimize a function $f$ on $\mathbb{R}^n$ and we apply Newton's method, updating: \begin{align} \mathbf{x}_{n+1} = \mathbf{x}_n - [\nabla^2 f(\mathbf{x}_n)]^{-1} \nabla ...
2
votes
0answers
94 views

Is there a generalization of eigenvalues/eigenvectors to $L^p$ for $p\neq2$?

Given symmetric positive definite $m\times m$ matrix $\Sigma$, $1 \le p,$ and constant $c > 0$, consider the solution to the optimization problem: $$ \min_{v : \left\|v\right\|_p \ge c} ...
11
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3answers
234 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
1
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0answers
26 views

Can we express a SPD matrix $S$ in terms of $S^{2}$ in a different manner to solve a convex problem?

I have to find the Symmetric Positive Definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$ which has been proven to be convex in the ...
1
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0answers
119 views

The double summation in the general quadratic form

According to my book and Wikipedia, a quadratic form on $\mathbb{R}^k$ is a real-valued function of the form $Q(x_1,...,x_k)=\sum_{i,j=1}^{k} a_{ij}x_ix_j.$ When I try to use this to check the general ...
0
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2answers
237 views

Non-degenerate quadratic form and non-singular matrix

Let $(V,Q)$ be a finite-dimensional quadratic space over a field $\mathbb{K}$. From my definition, $Q$ is non-degenerate if $\operatorname{rad}(V)=\{0\}$. How can I prove that $Q$ is non-degenerate ...
1
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2answers
135 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 ...
2
votes
2answers
58 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
934 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 ...
4
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0answers
116 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
327 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
262 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 ...
3
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2answers
378 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} ...
2
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4answers
343 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 ...
1
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4answers
316 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 ...
0
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1answer
94 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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votes
2answers
539 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
202 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: ...
7
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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
114 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
410 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
97 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 ...
16
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5answers
802 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
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2answers
97 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. ...
1
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1answer
252 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. ...
2
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2answers
610 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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0answers
185 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
828 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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0answers
38 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
91 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 ...
4
votes
1answer
164 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 ...
1
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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 ...
4
votes
1answer
259 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
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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
74 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
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3answers
195 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 ...
3
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3answers
218 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
136 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 ...