Questions about quadratic forms in multiple variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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10
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238 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 m^...
9
votes
0answers
91 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
7
votes
0answers
178 views

Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$ Set-Up: The type of Lie algebra of our interests (found out during a ...
6
votes
0answers
39 views

Why do isotropic spaces deserve their name?

Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions? From my experience so far, ...
6
votes
0answers
121 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 < ...
5
votes
0answers
107 views

Official name of Fermat's $x^2+3y^2$ theorem?

One of Fermat's more well-known claims is that for any prime number $p$, $p=x^2+3y^2\iff p\equiv 1\pmod 3$. Does this have an "official" name? (Another one which goes $p=x^2+y^2\iff p\equiv 1\pmod 4$ ...
5
votes
0answers
91 views

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $$\text{minimize} \quad x'Ax \qquad \qquad \text{subject to} \quad x'Bx = x'Cx = 1$$ Where $A$ is symmetric and $B$ and $C$ are diagonal. ...
5
votes
0answers
114 views

Distribution for ratio of dependent quadratic forms.

Random vector $\mathbf{x}_{0}$ $\sim$ $\mathcal{N}\left(\boldsymbol{\mu}, \mathbf{\Sigma} \right)$ is a sum of two orthogonal random vectors: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + $\mathbf{x}_{1}^{\...
5
votes
0answers
1k 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 ...
5
votes
0answers
176 views

Solutions to equation in matrix form

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
4
votes
0answers
51 views

The group defined by Gauss's definition of composition of forms

In article 242 of Disquisitiones, Gauss investigates the properties of the direct composition of two forms of the same discriminant. In this case, he gives a "natural" choice for such a composition. ...
4
votes
0answers
51 views

maximal linear subspaces contained in the cone over the Clifford torus.

Forgot: this is about Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0 I was a little surprised to find that, in the cone $x^2 + y^2 = z^2 + w^2$ in $\mathbb R^4,$ there are infinitely many ...
4
votes
0answers
292 views

Proving the max of a quadratic form ${\mathbf x}^T\mathbf A \mathbf x$ can be attained when $x$ is from $n$-dimensional hypercube

updated: Maybe my original question is somewhat misleading. I rewrite some of the post. This is some research problem I'm working on. I have an $n\times n$ symmetric positive-definite matrix $\...
4
votes
0answers
122 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 Z^{n\...
4
votes
0answers
437 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 ...
4
votes
0answers
181 views

Proof that the Arf invariant is independent of choice of basis

I'm confused about the proof of the following claim: Set $Z_2 = \mathbb{Z}/2\mathbb{Z} = \mathbb{F}_2$. Let $V$ be a $Z_2$-vector space of dimension $2n$ and let $e_i, f_i$ be a symplectic basis. Let ...
3
votes
0answers
24 views

Show $|E_1(\mathbb{F_q})|+|E_2(\mathbb{F_q})|=2(q+1)$

...under the assumption that $E_1,E_2$ are elliptic curves over $\mathbb{F_q}$ and that there is a (surjective) isogeny $\pi:E_1\rightarrow E_2$ defined over $\mathbb{F_{q^2}}$ obeying $\pi\phi_1=-\...
3
votes
0answers
36 views

Statement about composition of binary quadratic forms in “A Course in Computational Algebraic Number Theory”

On p.239 A Course in Computational Number Theory, Cohen writes "Although the group structure on ideal classes carries over only to classes of quadratic forms via the maps $\phi_{FI}$ and $\phi_{IF}$ ...
3
votes
0answers
43 views

Integral of a multivariate Gaussian distribution over quadratically separated partions

Imagine in the space of $\Re^n$, the quadratic curve $c: f(\mathbf{x}) = \mathbf{x}^TW\mathbf{x} + \mathbf{w}^T\mathbf{x} + w_0$ (with $W$ being a symmetric positive definite matrix, $\mathbf{w}$ a ...
3
votes
0answers
93 views

How can I find values for which a given expression gives a perfect square?

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
3
votes
0answers
52 views

Quadratic form and matrix

We know quadratic form $f(x_1,x_2)= a_{11} x_1^2 + 2 a_{12} x_1 x_2 + a_{22} x_2^2$ is non-negative for all $x_1,x_2 \in \mathbb{R}$ iff matrix $(a_{ij})_{2 \times 2}$ is semi-positive defined. My ...
3
votes
0answers
58 views

(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal matrix?...
3
votes
0answers
370 views

quadratic form from nxn matrices to reals ( Tr(A^2) ). I need to find it's signature and rank.

Firstly prove $Tr(A^2)$ defines a quadratic form from the space of $n \times n$ matrices to R. I think you just have to show that $Tr(A B)$ is a bilinear form which seems too easy to be correct or I'm ...
3
votes
0answers
152 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...
3
votes
0answers
87 views

Freeman Dyson's identity for the modular discriminant $\Delta$

In his 'Introduction to Modular Forms', Don Zagier states the Freeman Dyson's identity : $$\Delta(\tau)=\sum_{\substack{ (x_1,\ldots,x_5)\in \mathbb{Z}^5 \\ x_1+\cdots+x_5=0 \\ x_i \equiv i \...
3
votes
0answers
38 views

Algorithm for determining whether two real quadratic numbers are equivalent under a modular transformation

Let $\alpha \in \mathbb{C}$ be an algebraic number. If the minimal plynomial of $\alpha$ over $\mathbb{Q}$ has degree $2$, we say $\alpha$ is a quadratic number. Then $\alpha$ is a root of a unique ...
3
votes
0answers
80 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 ...
2
votes
0answers
59 views

On the integer solutions to $u^2+163v^2=w^3$ and others

It seems the solution of, $$u^2+dv^2 = w^3\tag1$$ involves the class number $h(d)$. Assume $\gcd(u,v)=1$. Q: For which $\color{red}{prime}\; d$ is the complete solution of $(1)$ in the integers ...
2
votes
0answers
51 views

Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$

Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$. Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
2
votes
0answers
27 views

$\det$ is the only multiplicative nonzero quadratic form on $\mathcal M_2(\Bbb R)$

Let $q$ a nonzero quadratic form on $\mathcal M_2(\Bbb R)$ verifying the relation $$\forall A,B\in\mathcal M_2(\Bbb R),\; q(AB)=q(A)q(B)$$ The question is to prove that $q=\det$. What I have tried ...
2
votes
0answers
46 views

Distribution of the norm of uniform random unit vector after linear transformation

Suppose that $\mathbf{u}$ is a uniform unit vector. It is obtained as $\mathbf{u}=\frac{\mathbf{n}}{||\mathbf{n}||}$ where $\mathbf{n}$ is a white Gaussian vector. Clearly we have $\mathbf{u}^T\mathbf{...
2
votes
0answers
39 views

A function of two quadratic forms

Given two functions $f(\mathbf{x^TAx})$ and $g(\mathbf{x^TBx})$, consider the new function $h(\mathbf{x})=f(\mathbf{x^TAx})g(\mathbf{x^TBx})$. Can $h$ be a function with a quadratic argument of the ...
2
votes
0answers
38 views

Looking for a proof of a known theorem involving integral quadratic forms

Let $n$ be a positive integer and let $Q$ be an integral quadratic form in $n$ variables. Let $M$ be the symmetric "two's in" matrix associated with $Q$ so that $Q$ can be expressed as the $1 \times 1$...
2
votes
0answers
35 views

Can systems of equations of diagonal quadratic forms be solved by Gaussian Elimination

Can the following system of equations be solved using Gaussian Elimination? $$ \begin{bmatrix} s_{00} & s_{01} & s_{02} & s_{03}\\ s_{10} & s_{11} & s_{12}...
2
votes
0answers
65 views

Trying to prove a theorem on simultaneous diagonalisation of matrices

Let $B$ be a $n\times n$ real symmetric positive definite form, and $A$ be a $n\times n$ real symmetric form. There exists an orthogonal matrix O such that $O^TBO=I$ and $O^TAO=\operatorname{diag}(\...
2
votes
0answers
19 views

Witt Groethendieck Ring splitting

I have a really basic question about the Witt Groethendieck ring of a field: In Lam's book, it says that $\hat{W}(F)/\hat{I}^2(F)$ depends only on the square classes of $F$, $\hat{W}/\hat{I}^2\cong \...
2
votes
0answers
62 views

Alternative way to recognize that a real symmetric quadratic form is positive

A real symmetric quadratic form $g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j$ is positive (definite) if $g(x)>0$ for every $\mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0$. It is well known that a ...
2
votes
0answers
32 views

Objects corresponding to Higher forms

If $Q$ is a quadratic form, then we know there exists matrix $A$ such that $Q=xAx'$ and $Q$ can be expressed as weighted sum of eigenvalues of $A$. If $H$ is a higher order form, then is there an ...
2
votes
0answers
49 views

Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$

Kindly see the question at the end of post. Solutions to the system of three equations, $$\begin{aligned} a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\ a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\ abcd\, ...
2
votes
0answers
40 views

can someone break this quad formula down for me?

Can someone explain how this person yield the stuff on the right side using quad formula?
2
votes
0answers
48 views

Classification of quadrtic forms over Q_p

I need some one to recap the topic with me and correct me when i am wrong. There are basically just a few questions at the end,but its important to also show what i know and not just what i dont. ...
2
votes
0answers
95 views

Law of large numbers for linear (quadratic) combinations of i.i.d. random variables

Let $(X_i)_{i\in\mathbb{N}}$ be i.i.d. real random variables with zero mean. By the law of large numbers $$\frac{1}{n}\sum_{i=1}^nX_i \to 0 \quad\text{(almost surely, in probabability...) as }\,n\to\...
2
votes
0answers
88 views

References for Composition Law on Binary Quadratic Forms

What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois Cohomology? It is my understanding that there is a cohomological approach, and I am studying ...
2
votes
0answers
60 views

Connection between class number and the theory of Ideals/Quadratic Fields

I've been studying the classic results in integer binary quadratic forms, mainly the equivalence and reduction of quadratic forms and the class number $H(d)$ (the definition I got for $H(d)$ is the ...
2
votes
0answers
60 views

Definition of the term 'generic' in context of quadratic forms.

In Proposition 3.3 of the paper: A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan Graphs, Combinatorica 8(1988), the authors use a result obtained by Malisev : "Let $f(x_1,\ldots,x_n)$ be a ...
2
votes
0answers
80 views

Classifing Second Degree Curves/Surfaces

I have got myself into a pickle with the following question: Classify the following (ellipse, hyperbola, ellipsoid etc) $x^2 + y^2 + 2z^2 + 2xz - 2y + 2z + 2 =0$ Now, I have written a symmetric ...
2
votes
0answers
44 views

Holzer reduction of solutions of quadratic ternary forms

Suppose $(x_{0}, y_{0}, z_{0})$ is a solution to the equation $ax^2 + by^2 + cz^2 = 0$. The solution is said to be Holzer reduced if $x_{0} < \sqrt{|bc|}$, $y_{0} < \sqrt{|ac|}$ and $z_{0} < \...
2
votes
0answers
42 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 ...
2
votes
0answers
100 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} v^{\top}\...
2
votes
0answers
138 views

Quadratic transformations of vector spaces

Much is known about transformations of the following form $$y_i = L_{ij}x_j \;\;: \;\; x\in\mathcal{R}^n, L\in\mathcal{R}^{n\times n}$$ We can infer a number of geometric properties about the ...