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
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1answer
46 views

Does an isotropic vector always exist for an indefinite quadratic forms?

I have some problem reading some paper. In that paper, the author proved that a quadratic form $Q$ has an isotropic vector(of course, nonzero) by showing $Q$ has both nonnegative and nonpositive ...
0
votes
0answers
31 views

Rank four quadratic form with trivial discriminant

Is there an example of a field $k$, quadratic form $\varphi$ of rank four, which is anisotropic over $k$, has trivial discriminant and is not a Pfister form? In case of rank six one can use Albert ...
3
votes
1answer
92 views

Solution to a System of Quadratic Equations

Problem: Solve for the values of a, b Equation 1: $$(x_1-a)^2+(y_1-b)=r^2$$ Equation 2: $$(x_2-a)^2+(y_2-b)^2=r^2$$ Where, $x_1, x_2, y_1, y_2$ and $r$ are all constant values For the ...
6
votes
1answer
129 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 ...
0
votes
0answers
16 views

example of quadratic form

Kindly give me an example of regular quadratic forms $q_1, q_2$ over a field $\mathbb F$ such that $D(q_1)=D(q_2)$ and $d(q_1)=d(q_2)$ but $q_1$ is not isometric to $q_2$, where $D(f)=\{d \in \mathbb ...
1
vote
1answer
95 views

Find max and min subject to constraint ||x|| = 4

$Q(x,y)=7x^{2}+12xy+12y^{2}$ I only know how to do this is $\|(x,y)\|=1$ If $\|(x,y)\|=1$, the eigenvalues are $16$ and $3$. So obviously $\min=3,\max=16$. I don't know what to do if ...
0
votes
2answers
40 views

Symmetric Matrix Quadratic Form

Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
3
votes
2answers
33 views

Rational quadratic forms

The quadratic form $$10x^2+20y^2+2z^2+4xy-6xz+8yz$$ can be written as $x^TAx$, where A = [ [10,2,-3] , [2,20,4] , [-3,4,2] ] Using diagonalization, this can be written in the form ...
1
vote
1answer
50 views

Projecting self-adjoint operator onto closed subspace

Let $H$ be a complex Hilbert space and let $(Q, D(Q))$ be a closed, densely defined, positive semidefinite, Hermitian quadratic form on $H$. (That is, $D(Q)$ is a dense subspace of $H$, $Q$ maps ...
2
votes
1answer
56 views

When does $x^TAx + c^Tx$ have a global minimum?

This question is closely related to my last question about extended quadratic forms. I figured out a nice criterion, when $$f : \mathbb R^n \rightarrow \mathbb R$$ $$f(x) = x^TAx + c^Tx$$ has a ...
1
vote
1answer
37 views

Rescaling of Ternary quadratic forms

I was reading about the Hilbert residue symbol, and the discussion of it starts out with the assumption that we can reformat any ternary quadratic form over the integers into the form ...
1
vote
3answers
136 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
1answer
75 views

Signed determinant of quadratic forms over Q_p

Let $W(k)$ be the Witt-Ring of the field $k$. in this script http://math.uga.edu/~pete/quadraticforms2.pdf at the bottom of page 2 the signed determinant is introduced by $d^\pm (q) = ...
3
votes
1answer
93 views

rank of quadrics

Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
1
vote
2answers
45 views

Quaternary Quadratic Forms

What is a Quaternary Quadratic Form? I've looked for a definition online and cannot find a precise clear definition. I am not taking a course. Just reading about quadratic forms. Thank you.
2
votes
1answer
59 views

numeric solutions on quadric surfaces

Maybe it's a trivial thing, but I can't seem to find solution I'm looking for. I need to find a parametric solution to the following equation ($\mathbf{A}$ is positive definite): $$ ...
1
vote
1answer
47 views

Primes of the form $x^2+ny^2$ where $n\equiv 1\pmod{4}$ is a squarefree number

Let $n\equiv1\pmod{4}$ be a squarefree number and $p\equiv1\pmod{4n}$ be a prime number. Does there exist $x,y\in\mathbb{N}$ such that $p=x^2+ny^2$?
0
votes
0answers
44 views

Quadratic reciprocity and Pfister forms

Let $p,q$ be different primes unequal to $2$. Let $(a/b)$ denote the Legendre Symbol. The following holds: $q\text{ is a square }\bmod p \Longleftrightarrow (q/p) = 1 \Longleftrightarrow X^2+qY^2 = ...
1
vote
1answer
40 views

How do i find a signature of a quadratic form? Also how do i represent a quadratic form as a sum/difference of squares?

For example given $(x,y,z,t) = xy+ y^2+ yz+z^2+zt$ How do i represent it as a sum and difference of squares (i.e. in the form $\sum a_iA_i^2$) and how do i find its trace? Or if i have a quadratic ...
0
votes
4answers
40 views

Finding the three unknowns

Can someone show me the steps to finding the three unknowns of these two equations. $$-a-bx+cx^2 = x^2+2x+1$$ The answers are $a=\ ...\ $, $b=\ ...\ $, and $c=\ ...$ , but I can't see how they ...
1
vote
3answers
64 views

How can I show the complete symmetric quadratic form has no zeros?

The quadratic complete symmetric homogeneous polynomial in $n$ variables $t_1,\ldots,t_n$ is defined to be $$h_2(t_1,\ldots,t_n) := \sum_{1 \leq j \leq k \leq n} t_j t_k = \sum_{j=1}^n t_j^2 + ...
1
vote
0answers
32 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. ...
3
votes
1answer
57 views

Question regarding quadratic form exercise in Hoffman Kunze

In the book the quadratic form associated with a bilinear form f is defined as $q(\alpha)=f(\alpha,\alpha)$. Then, if $U$ is a linear operator on $\mathbb R^2$ an operator $U^\dagger$ on the space of ...
1
vote
1answer
56 views

A particular quadratic minimization problem

Given $n^2$ constants $a_{11},a_{12},\ldots,a_{1n},a_{21},\ldots,a_{nn}$ and $n^2$ non-negative variables $x_{11},x_{12},\ldots,x_{1n},x_{21},\ldots,x_{nn}$. Find the minimum value of $$\sum_{i=1}^n ...
0
votes
0answers
28 views

Show that a quadratic form is bounded

I have a question (I hope I can formulate it adequately) concerning quadratic forms: If I have a quadratic form $$ T(x_1,x_2)=x^TQx, x=(x_1,x_2)^T $$ which is bounded to above, i.e. ...
0
votes
2answers
19 views

Positive definit quadratic form

I have a very elementar question... If I have a quadratic form $$ T(x_1,x_2)=x^TQx $$ and $Q$ is a positive definit symmetric 2x2-matrix, then does this mean that the quadratic form is positive ...
0
votes
1answer
19 views

Transformation of quadratic form

I've got the following quadratic form $T(x_1,x_2)=x^TQx$, with $$ x=\begin{pmatrix}x_1\\x_2\end{pmatrix}, Q=\begin{pmatrix}\frac{1}{2}(m_1+m_2)L_1^2 & \frac{1}{2} ...
2
votes
1answer
118 views

Quadratic Diophantine Equation $x^2 + axy + y^2 = z^2$

I have been reading about this quadratic Diophantine equation of the form $x^2 + axy + y^2 = z^2$ where x, y, z are integers to be solved and a is a given integer. All integral solutions are given ...
0
votes
0answers
36 views

On modular group and quadratic forms

Let $\Gamma$ be the modular group, is the group of linear transformations of the upper half of the complex plane. Let $\mathbb Q_{N^2{d_K}}/\Gamma$ (the group of positive definite primitive quadratic ...
0
votes
1answer
49 views

Proof of axis of symmetry [duplicate]

How do you prove -b/2a the Axis of symmetry equation using the Quadratic formula?
1
vote
1answer
94 views

Condition for the identity $(a-b)(a+b)=a^2-b^2$

The identity $(a-b)(a+b)=a^2-b^2$ holds true for what condition ???? I tried using real numbers and imaginary numbers but seems like it holds everywhere. Some say... only for ($a>b$) but I don't ...
0
votes
1answer
22 views

Expanding the Malhalanobis distance to find sufficient statistics of a multivariate Gaussian distribution.

Given a dat set $X=(x_1,...,x_N)^T$ in which the observations $x_n \in R^D$ are assumed to be drawn independently from a multivariate Gaussian distribution with mean $\mu \in R^D$ and covariance ...
0
votes
1answer
25 views

Reducible Quadratic form

What is a Reducible/Irreducible quadratic form?. I read about quadratic forms on wikipedia. I am reading this paper (A.O.L. Atkin, D.J. Bernstein, Prime sieves using binary quadratic forms, Math. ...
1
vote
0answers
43 views

Problem about $\mathbb{P}^3(K)$

Show that four skew lines in $\mathbb{P}^3$ have two transversals in common. I know that exist a quadric which contains three of the four lines....but i'm stuck EDIT: If the skew lines are ...
0
votes
0answers
26 views

constrained Quadratic forms

suppose I have the following quadratic form $$ {\bf{x^{\top}Ax}} = constant $$ where A is positive definite and I know that $$ {{\bf{x}}^ \top }{\bf{1}} = 1 $$ Is there an analytical solution to ...
1
vote
1answer
85 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 : ...
2
votes
1answer
90 views

Quadratic Form - New Axes = Eigenvectors of P, Order of Eigenvectors Important? [Kolman P539 Example 6]

Hypothesise that $P$ is the symmetric matrix of some quadratic form $g(\mathbf{ x} ) = \mathbf{ x^TAx} $. Then $P$ is the orthogonal matrix consisting of orthogonal eigenvectors of $A$. Moreover, use ...
0
votes
1answer
426 views

How to find quadratic function in vertex form from two points?

I'm starting to learn about quadratic formulas in math class. This question came up in a homework packet: A WNBA player takes a three-point shot 22 feet away from the basket, The ball reaches ...
2
votes
3answers
61 views

A is a matrix of positive defined quadratic form. How can I show, $A^{-1}$ is the same?

Let a square matrix A is a matrix of positive defined quadratic form. How can I show, that $A^{-1}$ also a matrix of a positive defined quadratic form? Positive defined quadratic form is A(x,y), that ...
0
votes
2answers
30 views

How to prove: a quadratic form with a matrix $ B = CC^T $ is positive defined?

Let a matrix $ C \in \Bbb K^{n \mathtt x n} : det(c) \ne 0 $ (K is any field - C or R) $ \Rightarrow $ a quadratic form with a matrix $ B = CC^T $ is positive defined one. How to prove it?
0
votes
1answer
42 views

If $4xy+3=c^2+3d^2$, is $xy$ necessarily a square?

I have a polynomial which, simplified, ends up in the form $$4xy+3 = c^2+3d^2.$$ Evidently $4xy+3$ is of the form $a^2+3b^2$, in light of the equality. But does $$ c^2 + 3d^2 = 4xy + 3 = xy(2)^2 ...
1
vote
1answer
38 views

Solving this equation

Question: Solve: $$3^{2x^2}-2\cdot3^{x^2+x+6}+3^{2(x+6)}=0$$ I thought that we can take $a=3^{x^2}$ and $b = 3^{x+6}$. Then equation becomes $a^2-2ab+b^2=0$, which obviously means $a-b=0$. ...
0
votes
0answers
21 views

Optimization with intervals

I am trying to solve a specific problem, and I was able to summarize it in the following optimization problem. I have a portfolio comprised of two assets. Asset 1 has return $r_1$, standard deviation ...
9
votes
3answers
412 views

A Pell equation inside a Pell equation

While working on another problem (see http://mathoverflow.net/questions/143599/solving-the-quartic-equation-r4-4r3s-6r2s2-4rs3-s4-1), I found the following equation to be solved: $$ ...
0
votes
2answers
29 views

Possibility of integral quadratic with these roots

If x and w are the roots of a quadratic equation with integral coefficients then is this possible: ${x = w = \frac{2}{3}}$. The correct answer says it is, but how is that so if it means: ...
1
vote
2answers
41 views

Finding matching roots

If ${4 + \sqrt{2}}$ is one root of a quadratic equation given by ${x^2 - Px + Q =0}$ where P and Q are rational numbers then find the missing root. The answer is ${4 - \sqrt{2}}$. And I'm a bit ...
3
votes
1answer
106 views

Applications of simultaneous diagonalization of quadratic forms

If $A$ and $B$ are square symmetric matrices and, additionally, one of them, say $B$, is positively defined, then there exists an invertible matrix $S$ such that $$S^{\top}\!AS=D ...
2
votes
1answer
57 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 ...
0
votes
1answer
42 views

Concavity of quadratic form

I know that the quadratic form $x'Ax$ is a concave in vector $x$ if matrix $A$ is negative semi definite. What happens if $A$ depends on $x$ (so that I have $x'A(x)x$), but I still know that $A(x)$ is ...
0
votes
0answers
15 views

Relaxed matrix factorization

I have an optimization problem like $ min~~ \frac{1}{2}||L||^2_F + \frac{1}{2}||R||^2_F,~~~~subject~to~ M=LR^T$ with respect to $L$ and $R$. I know that there are several factorizations that give ...