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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3
votes
2answers
123 views

Quadratic Equation with “0” coefficients

Let's say I have two objects $x$ and $y$ whose position at time $t$ is given by: $$ x = a_xt^2+b_xt+c_x \\ y= a_yt^2+b_yt+c_y $$ And I want to find which (if any) values of $t$ cause $x$ to equal ...
3
votes
1answer
90 views

Question about the definition of representability of a quadratic form

Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find ...
0
votes
3answers
118 views

$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form

Prove that every $n$-linear alternating form on a vector space of dimension less than $n$ is the zero form.
1
vote
1answer
50 views

$n$-linear form: An Interpretation

What is a good example of an $n$-linear form that is more familiar to a student learning at an elementary level? EDIT: I'm just trying to show that every $n$-linear alternating form on a vector ...
1
vote
1answer
45 views

How to show that $A=B-C$

How to show that for a real symmetric matrix $A,~A$ can be written as $A=B-C$ where $B,C$ are positive definite real symmetric matrices? Please help me ! I'm clueless.
1
vote
1answer
71 views

Solving quadratic form $\mathbf{x}^\mathrm{T}\mathbf{A}\mathbf{x} = c$ for $\mathbf{x}$

This is a simple question I hope, is there an easy way to solve: $$ \mathbf{x}^\mathrm{T}\mathbf{A}\mathbf{x} = c $$ for $\mathbf{x}$? (Assume $\mathbf{A}$ is positive definite). Geometrically the ...
1
vote
1answer
24 views

Eigenvalues of $\sum_{i=1}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}$

Consider the cuadratic form $$ \mathbf{x}^{\intercal}Q\mathbf{x} = \frac{x_1^2}{\lambda_1} + \sum_{i=2}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}\ . $$ Is it true that the eigenvalues of $Q$ are ...
6
votes
2answers
265 views

Coercive bilinear form on Hilbert space

I need to show the two following results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. Consider a continuous symmetric bilinear form $B$ on a ...
1
vote
1answer
159 views

prove that determinant is a quadratic form

let $V$ be a vector space of all $2 \times 2$ hermitian matrices with entries from $\mathbb C$, over the field $\mathbb R$. prove that $q(v)=\det(v)$ is a quadratic form. I tried to prove that ...
0
votes
1answer
30 views

Quadratic fit check

I've performed LS fit to data in order to fit the following quadratic function: $$f(x,y) = A~x^2 + B~y^2 + C~x~y + D~x+E~y +F$$ Now, I would like to check that the fitted function looks like a ...
1
vote
0answers
31 views

Solve I.V.P for differential using quadratic form

Solve the i.v.p for $y''+4y'+5y=0, y(\frac{\pi}{2})=1/2, y'(\frac{\pi}{2})=-2$ I solved using the quadratic form. and I got $\lambda = \frac{(-4 \pm 2i)}{2}$, which for $\lambda 1,2= 2+2i$. And then ...
1
vote
5answers
135 views

How do you determine whether the quadratic form is positive and negative definite?

How do you determine whether the quadratic form $Q(x,y) = 2x^2 - 4xy + 5y^2$ is positive definite, negative definite, or indefinite? Could someone show step by step with explanations? Thank you
6
votes
1answer
206 views

How to prove that $ E:=ABC D $ is also positive definite?

Now I think this is true: Let $A$, $B$, $C$ and $D$ be symmetric, positive definite matrices and suppose that $E:=ABCD $ is symmetric. How might I prove that $E$ is also positive definite? ...
0
votes
1answer
77 views

Generating vectors of the face-centered cubic lattice

I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by ...
9
votes
2answers
493 views

Generalizing the 290 theorem.

I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic ...
1
vote
1answer
736 views

Real and complex canonical forms of quadratic form

How do I find the canonical form of $$q_1(x,y,z)= 4x^2 +4xz+2yz$$ Now I have put it in matrix form as: $$\left( \begin{matrix} 4 & 0 & 2 \\ 0 & 0 & 1 \\ ...
1
vote
1answer
81 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 ...
2
votes
1answer
128 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
votes
2answers
150 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
vote
1answer
60 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 ...
-3
votes
2answers
139 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
votes
0answers
32 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
votes
6answers
375 views

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

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
0
votes
1answer
73 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
vote
1answer
150 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
vote
1answer
144 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
vote
3answers
82 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
106 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
91 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
votes
3answers
222 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
vote
0answers
24 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
vote
0answers
108 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
votes
2answers
182 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
vote
2answers
128 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
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 ...
1
vote
1answer
752 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 ...
3
votes
0answers
106 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
votes
0answers
276 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 ...
1
vote
0answers
234 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
votes
2answers
318 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
votes
4answers
278 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
vote
4answers
293 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
votes
1answer
90 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: ...
-2
votes
2answers
455 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 ...
0
votes
4answers
175 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
votes
2answers
973 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 ...
1
vote
0answers
91 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}\} \\\ ...
5
votes
1answer
326 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 ...
1
vote
0answers
91 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 ...
15
votes
5answers
706 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 ...