Questions about quadratic forms in multiple variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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73 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
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2answers
39 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 ...
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1answer
80 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 ...
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1answer
74 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 ...
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1answer
54 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 ...
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3answers
210 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 ...
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1answer
83 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) = ...
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1answer
190 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 ...
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2answers
68 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.
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1answer
83 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
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1answer
56 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 $p=x^2+ny^2$?
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1answer
53 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 = ...
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1answer
192 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 ...
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4answers
44 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 ...
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3answers
77 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 + ...
2
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0answers
48 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
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1answer
73 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 ...
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1answer
65 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 ...
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2answers
46 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
32 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
237 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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1answer
115 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
37 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
58 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
54 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
29 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
205 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
121 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
4k 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
73 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
38 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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2answers
49 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
40 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
26 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
521 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
31 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: ...
2
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2answers
57 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
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1answer
210 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
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0answers
91 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
49 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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2answers
61 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
27 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
53 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
78 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 ...
2
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1answer
99 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
45 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
54 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
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1answer
237 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
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1answer
45 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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2answers
42 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) $x^2$- ...