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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1answer
34 views

Condition on the positivity of a quadratic form

We place ourself in $\mathbb{R}^{n}$. Let's consider a positive definite matrix $M \in \mathcal{M}_{n} (\mathbb{R})$, $V$ and $E$ $\in \mathbb{R}^{n}$, and $\alpha > 0$. We consider the ...
1
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1answer
84 views

how to solve these two quadratic equations

Can someone help me find the solution for these two quadratic equations ? $ 2(z^2) \ - \ 3.023bz \ + \ 0.115(b^2) \ + \ 2.0814b \ + \ 0.142z \ - \ 0.5856 \ = \ 0 $ $ 6.0828(z^2) \ + \ 2.0414bz \ + \ ...
1
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0answers
26 views

Diagonalization of quadratic forms over $\mathbb{Q}$

I'm having difficulties in finding the diagonal forms of some quadratic forms. I am sure it is not supposed to be that difficult but I guess I am lacking some creativity after overdoze of coffee and ...
1
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2answers
74 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
1
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2answers
295 views

number of integral values for which $x^2+19x+92$ is a perfect square.

number of integral values of x for which $x^2+19x+92$ is a perfect square=? I have no idea how to do this. Please help.
4
votes
2answers
207 views

Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can ...
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2answers
146 views

steepest descent with quadratic form converge in 1 iteration

Well I'm stuck on an exercise given: The steepest descent method is applied to the quadratic form $$Q(\mathbf{x}) = \tfrac{1}{2}\mathbf{x}^TA\mathbf{x} - \mathbf{b}^T\mathbf{x} + c$$ where $A$, ...
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2answers
44 views

proving that $\max Q(x)=\lambda_\max$

Let $Q(x)$ be quadratic form. Prove that $\max_{\|x\|=1}Q(x)=\lambda_\max$. $Q$ is symmetric so it can be presented as $$\langle Ax,x\rangle$$ where $A$ is matrix which on its diagonal appears ...
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3answers
67 views

Please help me with a (simple?) “solve for x” problem.

I'm preparing for the GRE and was working through an old textbook (chapter on quadratic equations "completing the square," if that helps) and got stumped on $\displaystyle x^2 +{\frac{5x}{a}} + 6x^2 = ...
2
votes
1answer
104 views

Combining results with Chinese Remainder Theorem?

$9x^2 + 27x + 27 \equiv 0 \pmod{21}$ What is the "correct" way to solve this using the Chinese Remainder Theorem? How do I correctly solve this modulo $3$ and modulo $7$ without brute force?
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1answer
21 views

Positivity of homogeneous form of the fourth degree

I encountered an exercise that asked if the study of the positivity of $Q(u,v) = a_0 u^4 + a_1 u^3 v + a_2 u^2 v^2 + a_3 u v^3 + a_4 v^4$ can be reduced to the study of the corresponding problem ...
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2answers
75 views

a quadratic equation for two unknown number

find values of $p$ such that the equation $4x^2 + 3px - 2p = 0$ has? below are a few choices of the value p: a) 2 real roots b) 1 real roots c) no roots or complex roots so far i did for a) 2 ...
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1answer
122 views

completing the square for matrices

I'd like to calculate the posterior distribution given the prior distribution $w\sim N(0,\Sigma_p)$ and the likelihood $y|X,w\sim N(X^\top w,\sigma_n^2I).$ Ignoring everything that does not contain ...
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1answer
73 views

Counting function for sums of three squares

Legendre showed that an integer is the sum of three squares if and only if it is not of the form $4^n(8m + 7)$ for some nonnegative integers $n$ and $m$. However, I have been unable to find any ...
1
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1answer
149 views

Quadratic Map Solution

I read this on Wolfram Alpha. It states that: a quadratic recurrence relation uses a second degree polynomial to express $x_{n+1}$ as a function of $x_n$. A "quadratic map", then, is a recurrence ...
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1answer
35 views

Perform matrix-vector multiplicacion using quadratic form

I have a quadratic from, $v^{T}Hv +q^Tv+c$, where neither the $H$, $q$ and $c$ are given implicity. The whole expression is very complicated, but it is quadratic form. $H$ is simetric. The question ...
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1answer
43 views

Quadratic function question

Find an equation of the quadratic function whose f has zeros -1 and 3 and a maximum value of 8. I've tried to use intercept form, but I'm not sure wha to do with the maximum value of 8.
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1answer
135 views

Finding intersection points of 2 functions. My method is incomplete.

These are the 2 functions : $y = x^{4}-2x^{2}+1$ $y = 1-x^{2} $ Here's how I solved It : $x^{4}-2x^{2}+1 = 1-x^{2}$ $x^{4}-x^{2} = 0$ $x^2(x^2-1)=0$ $x^2-1=0$ $x=\pm \sqrt{1} $ Value of $y$ when ...
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0answers
132 views

Enumerating integer solutions to quadratic equations

Consider a quadratic equation with integer coefficients in two variables. $$ax^2+bxy+cy^2+dx+ey+f=0$$ I would like to know how to find the number of integer solutions $(x,y)$ to this equations. Is ...
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1answer
31 views

for each of the following solve for x and y

Question 1- For each of the following equations 1.1 Solve for x $$x^2-2xy+y^2=0$$ $$5x^2-3xy-8y^2$$ $$8x^2-5xy-13xy^2=0$$
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0answers
72 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 ...
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2answers
100 views

Linear Algebra Quadratic True False

"Every quadratic form $x^TAx$ with $A$ an invertible matrix is either positive definite, negative definite, or indefinite." Is this true or false? I am just wondering does it have to be positive, ...
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1answer
123 views

Signature of quadratic form and eigenvalues

I'm asking about the signature of the quadratic form - the triple (n0, n+, n−). Is it true that n+ is the number of positive eigenvalues, and n- is the number of negative of eigenvalues of the matrix ...
2
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1answer
95 views

Quadratic form and symmetric bi linear form formula, basic point unclear to me

Something really basic but I have to ask it: I was taught that the formula of symmetric bi linear form of the quadratic form if: $f(v,w) = 1/2(q(v+w)-q(v)-q(w))$ but $q$ is linear so what did we get ...
2
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0answers
56 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 ...
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2answers
170 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
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0answers
83 views

Does this quadratic form represent 1?

I am stuck on the following question in Lam's quadratic forms for a few days now. Let $a,b,c$ be three elements of a field $F$ such that $0 \neq a^2+b^2 \neq c^2$. Show that the quadratic form ...
3
votes
1answer
59 views

About the roots of a quadratic equation

Let $m_1$ and $m_2$ the real and diferent roots of the quadratic equation $ax^2+bx+c=0$. Do you know some way to write $m_1^k + m_2^k$ in a simplest form (linear, for example) using just $a,b,c,m_1$ ...
2
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0answers
50 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
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4answers
134 views

Find $\frac{a^3}{a^6 + 1}$ given a is a root of a quadratic equation

My question is: If a is a root of the equation $x^2 - 3x + 1 = 0$, then find the value of $\frac{a^3}{a^6 + 1}$. So, I figured we can use the Sridharacharya ...
4
votes
2answers
119 views

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$.

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$. Here $n\in\mathbb N$, $a,b,c,x,y,z\in\mathbb Z$. This problem is originally ...
2
votes
2answers
91 views

Is this expression a quadratic form

I have an matrix expression that basically is of the form: \begin{equation} tr(B X BX ) \end{equation} Where $B$ and $X$ and nonsquare matrices. $B$ is $p \times n$, $X$ is $n \times p$. It ...
0
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2answers
156 views

Congruent diagonal matrix

For two days I reflect on this question without an answer: If $A=(i+j-1)_{1\le i,j\le n}$ is matrix in $\mathcal M_n(\mathbb R)$, the question is to find basis in which $A$ is congruent to diagonal ...
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1answer
217 views

Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
0
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2answers
348 views

Find the symmetric matrix that represents the quadratic form $Q(X)=trace(X^2)$, $X\in mat_n\mathbb (R)$

as the title says, find the symmetric matrix (or signature) of $Q(X)=trace(X^2)$ where $X$ is an $n$ by $n$ matrix with real entries. the diagonal of $X^2$ is $$\sum_{k=1}^n x_{ik}x_{ki}$$ So ...
3
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1answer
122 views

Signature of quadratic form $Q(p)=p(1)p(2)+p(3)p(4)$

I was asked to find the signature of the quadtratic form $Q(p)=p(1)p(2)+p(3)p(4)$ where $p$ is a polynomial in $\mathbb R_n[x]$ I tried doing it via finding the symmetric matrix that $Q$ corresponds ...
0
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1answer
272 views

Find the signature of the quadratic form

Very simple question but something doesn't make sense to me. We are given a quadratic form (bilinear map but on the same vector twice): $Q(v) = v^t *\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 ...
1
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1answer
49 views

Showing that the quadratic form $Q(x,y,z,t)=x^2+y^2+z^2-7\cdot t^2$ is anisotropic on $\Bbb Q^4$

I'm looking for help in order to find a prove that the quadratic form $Q(x,y,z,t)=x^2+y^2+z^2-7\cdot t^2$ on $\mathbb Q^4$ can or cannot take the value $0$ on a nonzero element of $\Bbb Q^4$. I was ...
2
votes
2answers
199 views

Is it true that the whole space is the direct sum of a subspace and its orthogonal space?

Problem The ground field is $K$, $\operatorname{char}K\neq2$. Suppose $W$ is a (maybe infinite dimensional) subspace of a vector space $V$ with a symmetric/symplectic form ...
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1answer
128 views

Finding diagonal transformation matrix of a bilinear form

Let $f:\mathbb R^3 \times \mathbb R^3 \rightarrow \mathbb R$ be a symmetric bilinear form, and let $q$ be its quadric form, so that $q(x, y, z)= xy+yz$. Find the transformation matrix $A$ of $f$ by ...
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1answer
693 views

Polar form of quadratic equations

I'm trying to derive a polar, general and graphing, form of a quadratic equation. Here Is what I've done so far. $$ f(x)=ax^2+bx+c $$ And $$ f(x)=a(x-h)^2+k $$ Then I substituted $$ x=r\cos(\theta) ...
0
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2answers
64 views

Identify a quadric

Could you tell me how to identify a given quadric? Given a conic section, I should find an orthonormal affine frame in $\mathbb{R}^2$ (with standard dot product) in which the equation has a canonical ...
1
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1answer
63 views

Associated Bilinear Form to Q (Quadratic Form)

I need to diagonalize the quadratic form $Q(x) = {x_{1}}^{2} + 2x_{1}x_{2} + 2{x_{2}}^{2} + 2x_{2}x_{3} + {x_{3}}^{2}$ so I know I need to find the associated Bilinear form with $B(x,x) = Q(x)$ - the ...
0
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1answer
202 views

Quadratic formula / stationary points

For the question find and classify the stationary points of f(x) Given the function f(x) = ln(x^2 - 2x + 2) Are my calculations right in thinking x = 3.75, -0.75 ? Cheers
0
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1answer
67 views

Elliptical polarisation

In physic context one find the curve with parametrisation in t, $x=x_0\cos(t)$ and $y=y_0\cos(t+\varphi)$ with is an ellipse with equation ...
0
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0answers
204 views

selecting a good upper bound on quadratic form in presence of unknown PD matrix

I have a cost function that is \begin{equation} J=\text{Trace} [\ (\ I-LC)\ KQK^T(\ I-LC)\ ^T ]\ \end{equation} where $Q$ is a unknown positive definite matrix, $K$ and $C$ are full rank $n\times ...
0
votes
2answers
51 views

How I can find the matrix $A$ of this quadratic form?

Let $(e_1,\ldots,e_n)$ the standard basis of $\mathbb R^n$ and we consider the quadratic form $$\Phi(x)=\sum_{1\le i<j\le n}(x_i-x_j)^2$$ How I can find the matrix $A$ of this quadratic form? My ...
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0answers
109 views

How to solve an optimization problem with non-convex Frobenius norm constraint?

The form of my problem is: $$ \min_W \|Y-WX\|_F^2-\|V-WU\|_F^2 $$ $$ s.t. \|W\|_F=1 $$ All five variables are matrices. Since the norm constraint is a non-convex one, I have no idea how to solve this ...
4
votes
5answers
200 views

Looking for proof of no solution to 4-variable quadratic diophantine equation

Show there are no integers $a,b,c,d$ such that $$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$ Motivation: The ideal $I=(3,1+\sqrt{-17})$ in $R=\mathbb{Z}[\sqrt{-17}]$ has the ...
0
votes
1answer
59 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...