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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2answers
26 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
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1answer
40 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
32 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
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0answers
18 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
369 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
26 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
89 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
45 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
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1answer
38 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 ...
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0answers
13 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 ...
0
votes
2answers
27 views

What is the definition of a non-degenerate homogeneous quadratic form over a finite field?

I read in some finite geometry notes by S. Ball and Z. Weiner the following: A conic is a set of points of $PG(2,q)$ that are zeros of a non-degenerate homogeneous quadratic form (in $3$ ...
0
votes
1answer
19 views

Simplify quadratic polynomial with matrix

I am reading a paper and have trouble following equation (3): $$ (\mathbf{x}-\mathbf{d})^T \mathbf{A}_1 (\mathbf{x}-\mathbf{d}) + \mathbf{b}^T_1 (\mathbf{x}-\mathbf{d}) + c_1 = \\ \mathbf{x}^T ...
1
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0answers
25 views

Approximation of a quadratic form

Let $\mathbf{x}=(x_1,\cdots,x_n)^T\in\mathbb{R}^n$ and $A\in\mathbb{S}_{++}^n$ be a symmetric positive definite matrix. Also, let $Q\colon\mathbb{R}^n\to\mathbb{R}$ be the quadratic form given by $$ ...
3
votes
1answer
67 views

What's wrong with $\det(P) = -1$ : Change of variable for Quadric Forms ? [Kolman P552 8.7.25]

Would someone please explain "why $\det(P) = 1$ is required" and the general procedure of effecting this? Lay S7.2 didn't expound on this and neither does Kolman in S8.6-8.8. Identify the graph ...
5
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1answer
49 views

Why must P be orthonormal, and not just orthogonal, for change of variable in Quadratic Form? [Kolman P560 8.8.24]

Lay P402 : A change of variable is an equation of the form $x=Py$, where $P$ is an invertible matrix and $y$ is the (neW) coordinate vector of $x$ relative to the basis of $\mathbb{R}^{n}$ determined ...
0
votes
1answer
43 views

Find a relation between $a$ and $b$?

I would appreciate if somebody could help me with the following problem: Let $f(x)=x^2-2ax+b$, $a,b\in \mathbb{R}$ Q: Find a relation between $a$ and $b$ ? If $|x|\leq 1$ then $|f(x)|\leq1 $
1
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3answers
46 views

Quadratic equations and inequalites

For every positive integer $n$, prove that $$\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$$ Hence or otherwise, prove that $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$, where $[x]$ ...
0
votes
1answer
61 views

On representation of quadratic form

In linear algebra, a quadratic form is defined as $Q(x)=x^TAx$ for some (non-singular) matrix $A$ and any $x\in V$, where $V$ is a vector space. Actually, quadratic form can be any one satisfying ...
2
votes
1answer
36 views

Find two bilinear forms with the same quadratic form over $\mathbb F_2$

Let $V$ be a $K$-vectorspace with a bilinear form $\langle , \rangle$ and the associated quadratic form $q:V \to K, v \mapsto \langle v,v \rangle$. Let $K = \mathbb F_2$. Are there two different ...
0
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0answers
24 views

Stuck in quadratic forms and discriminats problem

So I'm stuck in a pretty easy question about discriminants and quadratic forms of equations. I have already proved one side of the problem: we suppose that $x_0, y_0$ are the solutions to the ...
-1
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2answers
27 views

Quadratic equation form?

Suppose we know that the sum of two positive numbers is $2k$ and their product is $m$ then which of the following will be its quadratic equation and why? 1) $x^2$+ $(2k)x$+ $m$= $0$ 2) $x^2$- ...
2
votes
2answers
94 views

how to find rational numbers satisfying the binary quadratic equation $x^2+3xy+5y^2=4$

I am looking for a generalisation of the solution of $x,y$ wich are rational numbers,they could be infinite,how can i find such solutions,integer solutions are obvious I have found that ...
0
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0answers
12 views

how to write a lattice $[\alpha,\beta ]$ in the form [$a,b+c\omega _7$]

$\fbox{1}$ if we write [$2-\sqrt{7},5+3\sqrt{7}$] in the form [ $a,b+c\omega _7$],what is the value of $a,b,c$ $\omega=\sqrt{7}$,since $ 7\equiv 3\mod 4$ $N(2-\sqrt{7})=4-7=-3$ $N( ...
0
votes
1answer
23 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
53 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 \ + \ ...
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0answers
15 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 ...
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2answers
66 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
106 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.
2
votes
2answers
71 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 ...
1
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2answers
50 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 $A$, ...
1
vote
2answers
36 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 ...
0
votes
3answers
64 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 = ...
0
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0answers
19 views

Why is it that quadratic forms seem fundamental for reciprocal or dual mappings?

In projective geometry mappings between points and hyperplanes (a "reciprocal" or "dual" mapping) often involve quadratic forms, e.g. in 3d projective space the polarity against an ellipsoid. The ...
2
votes
1answer
66 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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0answers
15 views

Positive definite integral quadratic form with minimal orthogonal group?

Are there explicit examples in every rank of positive definite integral quadratic forms with orthogonal group $\pm 1$?
0
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1answer
18 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 ...
1
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2answers
41 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 ...
1
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1answer
27 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 ...
0
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1answer
60 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 ...
0
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1answer
71 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 ...
1
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1answer
26 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 ...
0
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1answer
32 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.
0
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1answer
54 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 ...
0
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0answers
59 views

Finding all integral solutions of a positive definite quadratic equation

Let $q(x_1,\ldots,x_n)$ be an integral positive definite quadratic form. For $d\in\mathbb{N}$ the equation $$q(x_1,\ldots,x_n)=d$$ has a finite number of integral solutions. Is there an algorithm to ...
0
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0answers
54 views

a question about relationship between KKT matrix equation and optimal solution of quadratic problem.

I have a question regarding how the KKT matrix plays in solving for optimization problem: Is it correct that the optimal solution for quadratic optimization problem with positive definite hessian ...
1
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0answers
55 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 ...
0
votes
1answer
26 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
56 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 ...
1
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2answers
53 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, ...