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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6
votes
3answers
229 views

reference for linear algebra books that teach reverse Hermite method for symmetric matrices

January 13, 2016: book that does this mentioned in a question today, Linear Algebra Done Wrong by Sergei Treil. He calls it non-orthogonal diagonalization of a quadratic form, calls his first method ...
2
votes
2answers
165 views

Find the transitional matrix that would transform this form to a diagonal form.

Let the quadratic form $F(x,y,z)$ be given as below $F(x,y,z)=2x^2+3y^2+5z^2-xy-xz-yz$ Find the transitional matrix that would transform this form to a diagonal form. I got the symmetric ...
3
votes
3answers
1k views

How to solve inhomogeneous quadratic forms in integers?

If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ ...
13
votes
1answer
2k views

Proof of Legendre's theorem on the ternary quadratic form

Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff $\left(\frac{-bc}{a}\right)=\left(\frac{-ac}{b}\right)=\left(\...
3
votes
2answers
448 views

Isotropy over $p$-adic numbers

Over what $p$-adic fields $\mathbb{Q}_p$ is the form $\langle3, 7, -15\rangle$ isotropic?
1
vote
1answer
72 views

Small integral representation as $x^2-2y^2$ in Pell's equation

Let $k$ be a "representable" positive integer, in the sense that $k=|x^2-2y^2|$ for some integers $x,y$. Does it necessarily follow that $k$ can also be represented with small parameters, i.e. $k=|u^2-...
0
votes
1answer
93 views

Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 ...
3
votes
6answers
534 views

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

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
1
vote
3answers
121 views

Writing an expression as a sum of squares

I'd like to write $2xy+2xz+2yz$ in the form $a(\cdots)^2+b(\cdots)^2+c(\cdots)^2$ where each blank space is a linear combination of $x,y,z$. The closest I have is: $$(x+y+z)^2-(x-z)^2-y^2=2xy+4xz+2yz$...
12
votes
1answer
5k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
18
votes
5answers
1k 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 $\vec x$...
10
votes
0answers
231 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^...
1
vote
3answers
217 views

Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards
1
vote
3answers
265 views

Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution

Help solving over the integers: $$ax^2+by^2+cz^2+dxy+exz+fzy=0$$ where $(x_0,y_0,z_0)$ is a known integral solution and $a,b,c,d,e,f$ are integral coefficients. I found in Tito Piezas' identities the ...
0
votes
1answer
520 views

How to solve an equation of the form $ax^2 - by^2 + cx - dy + e =0$?

I am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions. Thanks!
0
votes
0answers
67 views

Simultaneously diagonalise two real quadratic forms

I would like to simultaneously diagonalise the quadratic forms $A=2x^2+3y^2+3z^2-2yz$, and $B=x^2+3y^2+3z^2+6xy+2yz-6zx$. Of course there's a theorem saying this is possible and I followed the ...
0
votes
1answer
54 views

Transforming quadratic forms, how is this theorem called?

In my textbook there is the following nameless theorem: Let $Q=\sum_{i,j=1}^n a_{ij}X_i X_j$ with $a_{ij}=a_{ji}\in K$ be a quadratic form in $n$ variables over a field $K$ not of characteristic $...
6
votes
4answers
191 views

Parametrization of $x^2+ay^2=z^k$, where $\gcd(x,y,z)=1$

$x,y,z$ be three coprime integers, $a \in \mathbb{Z}>0$ and $k$ an odd integer. How do I find all the non-trivial solutions of the diophantine equation? $$x^2+ay^2=z^k$$ Does the method which ...
1
vote
2answers
65 views

Using Lagrange's diagonalization on degenerate linear forms

Let $A=\begin{pmatrix}1 & 2 & 3\\ 2 & 3 & 4\\ 3 & 4 & 5 \end{pmatrix}$ be a real matrix. Find an invertible matrix $P\in M_{3}(\mathbb{R})$ such that $P^TAP$ is diagonal ...
1
vote
1answer
110 views

Elementary properties of integral binary quadratic forms

Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. $D = b^2 - 4ac$ is called the discriminant of $f$. We say $f$ is positive definite if $a \gt 0$ and $D \lt 0$(cf. this ...
12
votes
1answer
2k 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 ...
26
votes
2answers
857 views

Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?

We did in class $x^2+y^2$, which was easy, and we had for homework $2x^2+2xy+3y^2$, which I did (its values (if not square) must be divisible by form primes, or of the form $x^2+5y^2$, and clearly ...
10
votes
4answers
619 views

Representing a number as a sum of at most $k$ squares

Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to $$ n = x_1^2 + ...
4
votes
4answers
93 views

What are all the concordant forms $n$ such that $a^2+b^2 = c^2,\,a^2+nb^2=d^2$ for $n<1000$?

Part I. The list of congruent numbers $n<10^4$ such that the system, $$a^2-nb^2 = c^2$$ $$a^2+nb^2 = d^2$$ has a solution in the positive integers is known (A003273) $$n = 5, 6, 7, 13, 14, 15, ...
4
votes
2answers
221 views

Should isometries be linear?

Question Suppose $V$ is a (finite-dimensional) vector space over $F$ ($\operatorname{char }F\neq2$, due to user1551) equipped with a non-degenerate quadratic form $Q$, and $T$ is a distance-...
9
votes
2answers
2k 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 $...
2
votes
1answer
84 views

Transforming Diophantine quadratic equation to Pell's equation

I have been discussing the fastest and most efficient ways of solving QDEs in a separate question record (Alternative method to solve quadratic Diophantine equations). However, as suggested by individ,...
1
vote
3answers
3k 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 ...
1
vote
5answers
285 views

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

Let $\Gamma = SL_2(\mathbb{Z})$. Let $\mathcal{H} = \{z \in \mathbb{C}\ |\ \mathcal{Im}(z) > 0\}$ be the upper half complex plane. Let $\sigma = \left( \begin{array}{ccc} p & q \\ r & s \...
1
vote
7answers
480 views

positive definite quadratic form

Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite? Approach: The matrix of this quadratic form can be derived to be the following $$M := \begin{pmatrix} 1 & \...
6
votes
3answers
502 views

Etymology of the word “isotropic”

Given a quadratic form $q : V \rightarrow k$, a nonzero vector $v \in V$ is said to be isotropic if $q(v) = 0$. Any subspace of $V$ containing such a vector is also said to be isotropic, and the ...
1
vote
1answer
260 views

A condition for an odd prime to be represented by a binary quadratic form of a given discriminant

Let $f = ax^2 + bxy + cy^2$ be an integral binary quadratic form. We say $D = b^2 - 4ac$ is the discriminant of $f$. If $D < 0$ and $a > 0$, we say $f$ is positive definite. It is easy to see ...
1
vote
2answers
2k 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 $$ \mathbf{z'Az=z'Bz},~~...
1
vote
1answer
214 views

finding if two binary quadratic forms represent the same integers

I am currently working with quadratic forms for a given discriminant D; to get all primitive forms (one for each equivalence class) I found this website : http://www.numbertheory.org/php/classnoneg....
32
votes
2answers
1k views

Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = \...
9
votes
2answers
773 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 ...
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 ...
6
votes
1answer
120 views

On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$

The problem is to find four integers $a,b,c,d$ such that, $$a^4+b^4+(a+b)^4=2{x_1}^4\\a^4+c^4+(a+c)^4=2{x_2}^4\\a^4+d^4+(a+d)^4=2{\color{blue}{x_3}}^4\\b^4+c^4+(b+c)^4=2{x_4}^4\\b^4+d^4+(b+d)^4=2{x_5}...
5
votes
1answer
195 views

On products of ternary quadratic forms $\prod_{i=1}^3 (ax_i^2+by_i^2+cz_i^2) = ax_0^2+by_0^2+cz_0^2$

The equation, $$ (ax_1^2+by_1^2)(ax_2^2+by_2^2) = ax_0^2+by_0^2\tag1$$ has the well-known solution when $a=b=1$, $$ (x_1^2+y_1^2)(x_2^2+y_2^2) = (x_1 y_2 + x_2 y_1)^2 + (x_1 x_2 - y_1 y_2)^2$$ ...
3
votes
2answers
9k views

Find the EigenValues and EigenVectors of the matrix associated with quadratic form

Problem: Find the EigenValues and EigenVectors of the matrix associated with quadratic forms $2x^2+6y^2+2z^2+8xz$. I know how to convert a set of polynomial equations to a matrix but I have no clue ...
4
votes
1answer
97 views

What are numbers $n$ such that $a^2+nb^2 = c^2$ and $na^2+b^2 = d^2$?

Let $n$ and $a,b,c,d,$ be in the positive integers. I. For the system, $$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$ then $n$ is a congruent number. The sequence starts as $n=5,6,7,13,14,15,20,21,$ and so ...
3
votes
1answer
75 views

Pell's equation and representation elements of $\mathbb Z_p$.

We defined the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $c\not\equiv 0\pmod{p}$. Is it true that $f$ is onto?
2
votes
4answers
153 views

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
2
votes
2answers
97 views

On $p^2 + nq^2 = z^2,\;p^2 - nq^2 = t^2$ and the “congruent number problem”

(Much revised for brevity.) An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$x^3-n^2x = ...
1
vote
1answer
150 views

A problem on positive semi-definite quadratic forms/matrices

Suppose $a+b+c=0$ and (without loss of generality) $a\leq b\leq 0\leq c$, $a^2+b^2+c^2=1$, is the following quadratic form positive semi-definite? Thank you very much. \begin{equation*} \begin{split} ...
1
vote
4answers
646 views

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

Find the integer values for which $x^2+19x+92$ is a perfect square. Also, How to proceed if you have to find values ( not necessarily integer)?
8
votes
4answers
5k views

Determining if a quadratic is always positive

Is there a quick and systematic method to find out if a quadratic equation is always positive or may have positive and negative or always negative for all values to its variables? Say for a quadratic ...
6
votes
1answer
200 views

Expressing a quadratic form, $\mathbf{x}^TA\mathbf{x}$ in terms of $\lVert\mathbf{x}\rVert^2$, $A$

EDIT: This question is actually an attempt to solve this. Please take a look. Let $A$ be a symmetric postive-definite $n\times n$ matrix, i.e. $A\in\mathbb{S}_{++}^{n}$ Also, let $\mathbf{x}\in\...
4
votes
2answers
111 views

Proving the multiplicativity of a binary quadratic form

Consider the set $S$ of all integers of the form $x^2+y^2+4xy$, where $x$ and $y$ are integers. How could one prove the set $S$ is closed under multiplication? I have tried the bashy brute force ...
4
votes
0answers
80 views

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $\min_{x} (x'Ax)$ subject to $x'Bx = x'Cx = 1$. Where A is symmetric and B and C are diagonal. Does anyone have a suggestion for an ...