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
212 views

Should diffeomorphisms preserving arc length be affine?

Problem Suppose $\varphi\colon V=\mathbb R^n\to V$ be a differmorphism and $d\varphi$ is its tangent mapping. $\langle\circ,\circ\rangle$ is a nondegenerate (symmetric or symplectic) bilinear form on ...
5
votes
0answers
97 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: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + ...
3
votes
1answer
114 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
0
votes
2answers
36 views

Definite Quadratic Form

I got the following problem: Let $V$ be a real vector space and let $q: V \to \mathbb R$ be a real quadratic form, Prove that if the set $L = \{v \in V | q(v) \ge 0\}$ forms a subspace of $V$ then q ...
0
votes
1answer
119 views

Quadratic Forms and their Matrices.

1) How do you manage to transform a matrix from quadratic to canonical form? For instance, assume a linear transformation such that: $$Q(x,y,z)=x^2+2xz+z^2;$$ As far as I can see, in the ...
0
votes
2answers
85 views

Quadratic Equation to its binomial form

How do I convert a quadratic equation to its binomial form For example how does $x^2 - 12x - 13$ become $(x-13)(x+1)$ ?
1
vote
0answers
61 views

Integrate the ratio of quadratic forms

Please, help me to solve the folowing problem. Given two positive-definite $n$-dimensional matrices $A$ and $B$, need to integrate its ratio over unit ball: ...
0
votes
3answers
95 views

Additional solutions to quadratic equations which don't match the formula answer.

I'm hoping for link to some resource which can explain why the following is true. $$ x^2 + 104x - 896 = 0 $$ Using the quadratic formula we pull a = 1, b = 104, c = 896. Putting that into the ...
7
votes
2answers
289 views

How find this matrix $A=(\sqrt{i^2+j^2})$ eigenvalue

let the matrix $$A=(a_{ij})_{n\times n}$$ where $$a_{ij}=\sqrt{i^2+j^2}$$ Question: Find the difference $sign{(A)}$ can see this define:http://en.wikipedia.org/wiki/Sylvester's_law_of_inertia My ...
0
votes
1answer
143 views

Solution to a quadratic form

I'm trying to find a closed form solution of the following quadratic form for $x$. $x^{T}Dx = c$ where $c$ is just a constant placeholder for some terms on the other side. I know that, because $D$ ...
0
votes
0answers
96 views

How checking $(-1)^n$ for $H=H_{n+m}$ is equivalent to checking $(-1)^{m+1}$ for $H_{2m+1}$?

This is from "Mathematics from Economists" by Simon and Blume: To determine the definiteness of a quadratic form of $n$ variables, $Q(\mathbf{x})=\mathbf{x}^TA\mathbf{x},$ when restricted to a ...
1
vote
0answers
21 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
0
votes
1answer
53 views

Where does the “inertia” term come from? [duplicate]

Where does the "inertia" term in regard to quadratic forms (e.g. Sylvester's law of inertia) come from? Thank you!
0
votes
1answer
107 views

Is $B$ a positive or negative semidefinite?

Let $A$ be an $n\times n$ symmetric matrix. Then, $A$ is a positive semidefinite iff every principal minor of $A$ is $\geq0$; $A$ is a negative semidefinite iff every principal minor of odd order ...
3
votes
0answers
36 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 ...
1
vote
2answers
34 views

problem with quadratic equation two variable

I have following equation $a^2+4.8ab-b^2=0$ and I have problem with solving it, I don't know why $a=-5 $ or $ a=0.2 $
1
vote
3answers
318 views

How to put $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ in canonical form

We are given the equation $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ We did an example of this in class but the equation had less terms. I took a note in class that says : if there are linear terms, I have ...
0
votes
2answers
84 views

Binary quadratic forms whose discriminant is that of a quadratic number field

Let $K$ be a quadratic number field, $D$ its discriminant. Let $ax^2 + bxy + cy^2$ be an integral binary quadratic form such that $D = b^2 - 4ac$. It seems that gcd$(a, b, c) = 1$(see this question). ...
0
votes
1answer
536 views

Quadratic form in canonical form

Reduce the quadratic form $q(x,y) = 6xy$ using the orthogonal reduction (i.e, find a orthogonal basis such that the matrix of the bilinear form is diagonal and $a_{ii} = 0$ or $a_{ii} = ^+_-1$) What ...
0
votes
1answer
124 views

Hyperbolic lattice and its cone

By lattice we mean a finitely generated free abelian group $L$ equipped with an integral non-degenerate symmetric bilinear form $L\times L\rightarrow\Bbb{Z}, \ (x,y)\mapsto x\cdot y$. We call $L$ ...
2
votes
0answers
59 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 ...
1
vote
1answer
52 views

Every 4-dimensional isotropic quadratic space of determinant 1 is hyperbolic

There is this claim in Scharlau's "Quadratic and Hermitian forms", Every 4-dimensional isotropic quadratic space of determinant 1 is hyperbolic. How can we prove it? I know that any ...
0
votes
3answers
125 views

What integers can be represented by the quadratic form $4x^2 - 3y^2 - z^2$?

Actually, I need to find if $4x^2 - 3y^2 - z^2 = 12$ is solvable. But I somehow feel that applying theory of integer representation by quadratic forms in three variables would yield quicker results... ...
-1
votes
1answer
76 views

Algorithm for finding full representatives of the orbit space of imaginary quadratic numbers of discriminant $D$ under the modular group

Let $\Gamma = SL_2(\mathbb{Z})$. Let $\mathcal{H} = \{z \in \mathbb{C}\ |\ Im(z) > 0\}$ be the upper half plane of complex numbers. Let $\sigma = \left( \begin{array}{ccc} p & q \\ r & s ...
1
vote
5answers
277 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 ...
0
votes
1answer
59 views

if f(x) is the polynomial (coeff of leadin term is unity) in 'x' of least degree such that f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1, then f(0)=?

If $f(x)$ is the polynomial (coefficient of leading term is unity) in 'x' of least degree such that $f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1$ Then $f(0)= ?$
4
votes
2answers
167 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 ...
1
vote
1answer
102 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 ...
1
vote
0answers
111 views

How to compute the class group of an order of a quadratic number field

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, i.e. the subring of $K$ which is a free $\mathbb{Z}$-module of rank $2$.Let $D$ be its discriminant. We use the notation and the result ...
1
vote
2answers
95 views

independent chi squares mean independent non central chi square?

Let $Y$ be a multivariate normal random vector with covariance $\Sigma$. Let $A_0,A_1$ be matrices such that $$A_0\Sigma A_1=0.$$ It is known that in this case $Y'A_0Y$ and $Y'A_1Y$ are independent ...
0
votes
2answers
575 views

Conditions for a real binary quadratic form to be positive definite

Since this question was heavily downvoted, I would like to change the presentation of the question as follows. I hope those of you who downvoted this question would be satisfied with the change. In ...
1
vote
0answers
118 views

Please help in solving $ax^2 + bxy + cx + dy + e$ = 0

Sometime back when trying to work out how to solve $ax^2 - by^2 + cx - dy + e = 0$ I learned that the way to solve such forms is to 'square the terms' and give it the form $A^2 - B^2 - E = 0$, $A = ax ...
1
vote
1answer
23 views

Explicit Isomorphism between $O(3,3)$ and $GL(4, \mathbb{R})$

I have seen it stated that $O(3,3) \cong GL(4, \mathbb{R})$, but I have never seen the isomorphism explicitly defined. Does anyone know what the isomorphism is or where I might be able to find it? ...
0
votes
2answers
42 views

From the quadtratic form of a matrix to its symmetric matrix

I am trying to solve this quadratic form: $$ f(x_1 , x_2 , x_3) = x_1^2 + x_2^2 + 5x_3^2 -2x_1x_2 + 6x_1x_2+ 3x_2x_3$$ I know that the quadratic form is defined as: $$f(x)= x^t Qx$$ However my ...
-1
votes
1answer
163 views

Relation between an integer represented by a binary quadratic form and a certain Dirichlet character defined by Jacobi symbol

Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $f$. It's easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). If ...
2
votes
1answer
215 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 ...
0
votes
1answer
256 views

how to simplify a general plane conic section's equation by linear algebra?

When encountering a general plane conic section a11x^2+a12xy+a22y^2+b1x+b2y+c=0, i can write it in matrix form as a quadratic form of the vector [x,y,1]. by what then? what should be done to reach the ...
1
vote
0answers
39 views

System of symmetric quadratic equations

Suppose $A_1, A_2, \ldots A_k$ are real symmetric (but possibly singular or indefinite) matrices. I want to know whether the system of quadratic equations $$v^T A_i v =0 $$ has a nontrivial solution ...
0
votes
1answer
41 views

Derivative of quadratic form of matrix in terms of the matrix elements?

Suppose I have $b^tAc$ and I try to get the derivative in terms of $A$. How could What is the matrix notational result? I believe the answer is $bc^t$, isn't it ?
0
votes
1answer
70 views

What is the significance of using variables h and k in vector form?

I'm just curious what the historical significance of using h,k in vector form are. It's very likely that the answer is, there is no significance just like there is no significance to using x,y,z. ...
5
votes
2answers
151 views

Why are quadratic forms so special and why not investigate in higher forms?

Ok, this is a soft question. If $K$ is a field of characteristic different from $2$, one can use the polarization identity to get a one-to-one correspondence between homogeneous polynomials of ...
2
votes
3answers
77 views

What finite fields are quadratically closed?

A field is quadratically closed if each of its elements is a square. The field $\mathbb{F}_2$ with two elelemts is obviously quadratically closed. However, testing some more finite fields on this ...
2
votes
3answers
53 views

Number of values of x

$$a\dfrac{(x-b)(x-c)}{(a-b)(a-c)}+b\dfrac{(x-c)(x-a)}{(b-c)(b-a)}+c\dfrac{(x-a)(x-b)}{(c-a)(c-b)}=x$$ How many values of $x$ satisfy this equation? It is clear that x=a, x=b, x=c do satisfy the ...
0
votes
1answer
61 views

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$.

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and either of the following is true: $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$. We see that if $(a,b,c)$ is a solution, then so is ...
0
votes
1answer
96 views

What is the geometric interpretation of quadratic forms?

I am trying to make sense of the following condition: Let $w_1, \dots, w_m \in \mathbb{C}^d$ with $\|w_i\| \le 1$ and $\sum_{i = 1}^m \, |\langle u, w_i \rangle |^2 = n$ for some $n \in \mathbb{R}$ ...
1
vote
3answers
65 views

What restrictions are on th sum of two fourth powers?

I've got an equation of the form $$ a^4+1=2b. \qquad(\star) $$ By well-known results regarding the sum of two squares, $b$ must be the sum of two squares. But does $(\star)$ force any other ...
0
votes
2answers
235 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
1answer
79 views

bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
-1
votes
1answer
79 views

The inverse class of the class represented by a primitive binary quadratic form of discriminant $D$

We use the definitions of this question. Is the following proposition true? If yes, how do we prove it? Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ ...
0
votes
1answer
37 views

Prove: matrix of the quadratic form is adj(A)

Suppose $A=\left(a_{i j}\right)_{n\times n}$ is an invertible real symmetric matrix. Prove: matrix of the quadratic form \begin{align}f\left(x_1,\text{...},x_n\right)=\left| \begin{array}{cccc} 0 ...