Questions about quadratic forms in many variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.
1
vote
0answers
14 views
Convergence in distribution of a quadratic form
If $Q_n=X_nM_nX_n=\sum_{i,j=1}^n X_i m_{nij}X_j$, $X_n=(X_1,...,X_n)$ where $X_j$ are iid random variables and $M_n=(m_{nij})$ is a symmetric matrix with extending rownumber in $n\to\infty$.
Iam ...
2
votes
1answer
16 views
Quadratic form $\mathbb{R}^n$ homogeneous polynomial degree $2$
Could you help me with the following problem?
My definition of a quadratic form is: it is a mapping $h: \ V \rightarrow \mathbb{R}$ such that there exists a bilinear form $\varphi: \ V \times V ...
0
votes
1answer
27 views
Matrix of quadratic form has to be symmetric?
On Wikipedia it is stated that any $n\times n$ real symmetric matrix A determines a quadratic form. But isn't $ax^2 + bxy + cxy + dy^2$, the quadratic form given by $v^T A v$ with $A=\begin{bmatrix}a ...
0
votes
0answers
22 views
Tensor product with $\mathbb{R}$ of an even unimodular lattice
Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$.
By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyperbolic plane.
Now ...
0
votes
0answers
20 views
Positive subspaces of quadratic forms
here's my question:
Let $V$ be a k-dimensional vector space over $\mathbb{R}$ and $q$ a quadratic form on $V$ of signature $(m,n)$ , $m+n=k$.
We have $W\subset V$ a positive (with respect to the ...
3
votes
1answer
32 views
Number of solutions of a positive integral quadratic form is finite?
Is there an easy way to see the following:
Suppose Q is an integral quadratic form in $n$ variables that is positive definite, that is $Q(x) \geq 1$ for all $0 \neq x \in \mathbb{Z}^n$. Then the ...
3
votes
1answer
75 views
Any integer can be written as $x^2+4y^2$
If $n$ is a positive integer with $(n,8)=1$ and $-4$ is square $mod$ $n$ then $n$ can be written in this form:
$n=x^2+4y^2$.
I was using that there are x, y integers satisfying $x^2+4y^2=kn$ where ...
0
votes
1answer
34 views
Real part of quadratic form
Suppose $q$ is a quadratic form on $\mathbb{C}^n$: $q(x)=x^HAx$, with $H$ denoting the hermitian transpose. Since I am only interested in the real part of $q$, I am trying to determine a matrix $B$ so ...
2
votes
1answer
47 views
Diagonalising quadratic form
Given the quadratic form $$Q(x) = \alpha\alpha_1\alpha_2 + 2\alpha^2\alpha_1\alpha_3$$ on $\mathbb{R}^2$ where $x = (\alpha_1,\alpha_2,\alpha_3)$ in some basis I want to find the signature of $Q$ ...
1
vote
2answers
34 views
Definitions and questions related to projective space $\mathbb{R}P^3$
I have the following questions regarding the definition of a quadric in a real projective space.
What is the precise definition on a quadric of signature (1,1) in the projective space ...
2
votes
1answer
42 views
What's a BETTER way to see the Gauss's composition law for binary quadratic forms?
There is a group structure of binary quadratic forms of given discriminant $d$:
Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$
The composition of two binary quadratic ...
0
votes
2answers
42 views
Difficulty in Quadratic equation and realtion with irrational roots
One root of the quadratic equation $ax^2 +bx + c=0$ is $\dfrac{2}{\sqrt{3} + \sqrt{5}}$. If $\frac{c}{a}$ is rational, then how do we find the other root. the answer given is that the other root is ...
0
votes
0answers
43 views
Solving for a sum of products
Following a question I asked yesterday, which yielded little success, I've refined my problem further to solving a system of equations.
In essence, I wish to solve for $h_n$ and $\hat{h}_n$ (for all ...
0
votes
0answers
34 views
Writing a quadratic form as a sum of squares
Let $Q(x_1,x_2, \ldots ,x_n)$ be a positive definite real quadratic form in the variables $x_1, \ldots ,x_n$. It is not hard to see that the function $f(x_1, x_2, \ldots ,x_n)=\frac{Q(x_1,x_2, \ldots ...
0
votes
1answer
25 views
Problem on hyperbolic hyperboloid generated by a rotation
This is the problem:
In $\mathbb{E}^3$ we consider the conic $\gamma$ of equations $x=yz-2=0$ , the line $a$ of equations $x=y+z=0$ and the surface $Q$, that is generated by the rotation of $\gamma$ ...
1
vote
1answer
20 views
solving for one variable in terms of others
A question from Steward's Precalculus textbook 5th, Pg 55,
the original formula is $$h=\frac{1}{2}gt^2+V_0t$$
the question asks to write the formula in terms of $t$, the answer is ...
3
votes
1answer
52 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 ...
2
votes
1answer
37 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
31 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
43 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
60 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
23 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
74 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
70 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
25 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
26 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
50 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
0
votes
1answer
37 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 ...
5
votes
1answer
132 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
143 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:
$$\left(
\begin{matrix}
4 & 0 & 2 \\
0 & 0 & 1 \\
...
1
vote
1answer
37 views
Quadratic Forms in Non-Linear Optimization
This is a rather trivial question but I am having a great deal of trouble:
Let $f(x) = (1/2)xQx-xb$
and $E(x) = (1/2)(x-x^*)Q(x-x^*)$
then $E(x) = f(x) + (1/2)x^*Qx^*$
where $x,x^*,b$ are vectors ...
2
votes
1answer
41 views
In quadratic form, how would symmetric matrix $A$ would change under coordinate change?
In http://en.wikipedia.org/wiki/Quadratic_form,
Let $q$ be a quadratic form defined on an n-dimensional real vector
space. Let $A$ be the matrix of the quadratic form $q$ in a given
basis. ...
1
vote
2answers
26 views
Finding a parametrization of a hyperbola who has a fixed signature
How do I find parametrization of the hyperbola $x^2-y^2=1$ which is the unit sphere of a quadratic form with signature $(1,-1)?$
The only parametrization that comes to mind is $x=\cosh t,y=\sinh t$. ...
1
vote
1answer
22 views
Determining a norm from a quadratic form
If $B$ is a quadratic form over some space $V$, what is the norm determined by $B$? Is this the inner product $\langle Bu,Bv\rangle$?
If not, and it is not possible to determine a norm from knowing ...
-3
votes
2answers
114 views
Lucky Lattice Points
How many lattice points lie on the sphere given by following equation ?
$$x^2+y^2+z^2=2013$$
Hint:
A lattice point has integer coordinates.
2
votes
0answers
28 views
$U$ is isotropic $\implies$ $U\subset U^0$
Let $(V,Q)$ be a quadratic module and $U$ is a subspace of $V$. Serre (A Course in Arithmetic, p. 29) claimed that the following is evident:
$U$ is isotropic $\iff$ $U\subset U^0$.
In other ...
3
votes
4answers
156 views
Integral solutions of hyperboloid $x^2+y^2-z^2=1$
Are there integral solutions to the equation $x^2+y^2-z^2=1$?
0
votes
1answer
50 views
Rewriting a quadratic Matrix equation as a quadratic vector equation
Consider the set of $N \times N$ matrices $\{W_i\}_{i=1}^{i=L}$, set of $N \times 1$vectors $\{g_i\}_{i=1}^{i=L}$ and $\{h_i\}_{i=1}^{i=L}$. Now consider the following sum
\begin{align}
...
1
vote
1answer
55 views
Find a matrix that simultaneously diagonalizes to matrices
struggling with a question from homework and would appreciate some assistance.
Let $A, B \in M_2^{\mathbb{R}}$ be defined as follows:
$$A = \begin{pmatrix} 2 & 1 \\ 1 & 1 ...
0
votes
1answer
66 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
53 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
66 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
68 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
139 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
16 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
46 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
71 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
100 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
52 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 ...
