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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26 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. ...
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2answers
15 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 ...
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1answer
17 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
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1answer
97 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 ...
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0answers
25 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 ...
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1answer
35 views

Proof of axis of symmetry [duplicate]

How do you prove -b/2a the Axis of symmetry equation using the Quadratic formula?
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1answer
88 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 ...
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1answer
19 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 ...
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1answer
20 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. ...
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0answers
42 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 ...
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0answers
24 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 ...
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1answer
48 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 : ...
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1answer
78 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 ...
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1answer
81 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 ...
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3answers
48 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 ...
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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?
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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 ...
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1answer
35 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 $a-b=0$. ...
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0answers
19 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
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3answers
383 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: $$ ...
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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: ...
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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 ...
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1answer
91 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 ...
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1answer
46 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 ...
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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
14 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 ...
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2answers
30 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 $3$ ...
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1answer
21 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 ...
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0answers
28 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 $$ ...
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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 ...
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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 ...
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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 $
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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 $[x]$ ...
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1answer
68 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 ...
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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 ...
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0answers
25 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 ...
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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) $x^2$- ...
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2answers
101 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 ...
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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( ...
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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 ...
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1answer
55 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
68 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} ...
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2answers
125 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
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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 ...
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2answers
54 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
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 ...
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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 = ...
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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 ...
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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?