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
63 views

The inverse class of the class represented by a primitive binary quadratic form of discriminant $D$

We use the definitions of this question. Is the following proposition true? If yes, how do we prove it? Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ ...
0
votes
1answer
33 views

Prove: matrix of the quadratic form is adj(A)

Suppose $A=\left(a_{i j}\right)_{n\times n}$ is an invertible real symmetric matrix. Prove: matrix of the quadratic form \begin{align}f\left(x_1,\text{...},x_n\right)=\left| \begin{array}{cccc} 0 ...
-1
votes
1answer
62 views

Standardize a Quadratic Form

standardize Quadratic Form $$8x_1x_4+2x_3x_4+2x_2x_3+8x_2x_4$$ how to do? what's the simplest method. How to choose the first linear replacement, matrix or do ...
1
vote
1answer
131 views

Multivariable local maximum proof

Suppose we have a twice differentiable function $f: \mathbb{R} ^n \to \mathbb{R}$, a point ${\bf x^0} = (x_1 ^0 , \ldots , x_n ^0)$ and we know that $\nabla f({\bf x}^0) = 0$ $({\bf x - x^0})H({\bf ...
1
vote
0answers
80 views

About quadratic form and its discriminant

There are 3 parts of the problem. Let d be a perfect square, possibly 0. Show that there is a quadratic form $ax^2+bxy+cy^2=0$ of discriminant d for which a=0. Let a,b,c be integers with $a\ne0$. ...
1
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0answers
44 views

Witt index and Pfister neighbours

Suppose $\phi$ be a 8-dimensional quadratic form with trivial discriminant over a field $F$ of characteristic not 2. Assume that there is 3-fold Pfister form $<<a,b,c>>$ such that ...
1
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4answers
538 views

Real world examples of quadratic and/or finding roots of a quadratic?

Anyone ever come across a good situation where a) a situation is modeled by a quadratic equation $y=ax^2+bx+c$ and/or b) you've even needed to find where $y=0$ (roots, $x$-intercept, etc)
1
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1answer
42 views

Quadratic Form $f\left(x_1,x_2,\cdots ,x_n\right)=\sum _{i=1}^m \left(a_{i 1}x_1+\cdots +a_{i n}x_n\right)^2$

Quadratic Form $f\left(x_1,x_2,\cdots ,x_n\right)=\sum _{i=1}^m \left(a_{i 1}x_1+\cdots +a_{i n}x_n\right)^2$, i) write the corresponding matrix; ii) when $a_{\text{ij}}$ are all real numbers, gives ...
0
votes
2answers
69 views

Integer roots of $x^2+y^2 = 25$ and $x^3+y^4=145$

I'm trying to find the solution of $x^2+y^2=25$ and $x^3+y^4=145$. I tried doing substitution, which leads me to: $y^2=25-x^2$ substituted to $x^3+y^4=145$ $x^3+(25-x^2)^2=145$ ...
1
vote
3answers
131 views

A problem about the quadratic form $x^TAx=0$.

$A$ is an $m\times m$ real matrix in $\mathbb{R}^{m\times m}$. If $x^TAx=0\ \forall\ x\in\mathbb{R}^m$, can we conclude that $A=0_{m\times m}$? Why? Note: $x^T$ is the transpose of $x$. $0_{m\times ...
0
votes
0answers
37 views

Solving intersection of 4 quadratic equation with constraints

I want to find the intersection of 4 quadratic equation with the constraint that $\left[\begin{matrix}q_0 \\ q_1 \\ q_2 \\ q_3 \end{matrix} \right]$,$\left[\begin{matrix}p_0 \\ p_1 \\ p_2 \\ p_3 ...
0
votes
1answer
71 views

Extremum of a multidimensional quadratic function

I have the following function: $$ g(h) = h'\Sigma\Sigma'h-h'm-r, $$ where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
2
votes
1answer
74 views

A problem about the discrete logarithm

suppose there are a multiplicative cyclic group $F_p^*(p \;is\;big\; prime)$, and $G=\langle g \rangle(g \;is\; a\; generator)$ is a subgroup of it and $G$'s order is $q(q\;is\;big\;prime \;and ...
0
votes
0answers
85 views

Finding the number of integral values of a for which the inequality $3- |{x-a}|>x^2$ is satisfied by at least one negative value of $x$.

Finding the number of integral values of a for which the inequality$$ 3- |{x-a}|>x^2$$ is satisfied by at least one negative value of $x$.Here's a duplicate post ->{{Determine all the values of ...
1
vote
1answer
192 views

How high is a baseball after 5 seconds of being thrown off of a 175 foot building? And how many seconds does it take to hit the ground? [closed]

A person standing close to the edge on the top of a 175-foot building throws a baseball vertically upward. The quadratic equation $$h = -16 t^2 + 160 t + 175$$ models the ball's height, $h$ , above ...
1
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4answers
295 views

How can I find integer values for which a given expression gives a perfect square?

Find the integer values for which $x^2+19x+92$ is a perfect square. Also, How to proceed if you have to find values ( not necessarily integer)?
4
votes
0answers
154 views

Proving the max of a quadratic form ${\mathbf x}^T\mathbf A \mathbf x$ can be attained when $x$ is from $n$-dimensional hypercube

updated: Maybe my original question is somewhat misleading. I rewrite some of the post. This is some research problem I'm working on. I have an $n\times n$ symmetric positive-definite matrix ...
0
votes
1answer
169 views

Ratio of dependent chi squared random variables

Suppose that $X=v'A_1v$ and $Y=v'A_2v$, where $A_i$ are symmetric matrices and $v$ a multivariate normal vector with covariance $V$, are chi squared distributed each with its own degrees of freedom. ...
2
votes
1answer
72 views

finding zeroes of a quadratic form

Let $a,b\in\mathbb Z$ be squarefree with $a>0$. Suppose that I know that there exist $(0,0,0)\neq (x,y,z) \in \mathbb Z^3$ s.t. $x^2-by^2-az^2=0$. Is there any known algorithm to find any such a ...
2
votes
1answer
98 views

Graphing quadratic form, which eigenvalue should be chosen first?

I just graph a quadratic function, $-4x^2_1+4x_1x_2-7x_2^2=-8$, by: Find the eigenvalues of the function above, which are $\lambda_1=-8$ and $\lambda_2=-3$ Use the eigenvalues to make a new ...
2
votes
1answer
72 views

Are $Q_1(x,y)=xy$ and $Q_2(x,y)=x^2+y^2$ equivalent forms?

This is an old qual problem at my school. I've never dealt with quadratic forms, so I'm not sure if I've done this right. Two quadratic forms $Q_1(x,y)$ and $Q_2(x',y')$ are said to be equivalent ...
1
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1answer
39 views

Showing a particular function on the space of polynomials w/ degree $ \le 2$ is a quadratic form, and computing signature

I've been a long way from linear algebra but I have to go back to it for an exam, and I've found myself stuck on the following question. Define $Q$ on the space of all polynomials with degree at most ...
4
votes
2answers
365 views

Clifford Algebras

What would be the best source to learn Clifford Algebras from? Anything online would suffice or any textual sources for that matter.. I'm interested in doing a project in the subject, but I'm not ...
0
votes
1answer
65 views

How do i solve this square root to solve for $y$ and get rid of $x$ at the same time

This is what I have and I don't know how to solve for $y$. Because $x$ is a variable it would be great if it would cancel out. $$ y = \frac{\sqrt{2.25^2Ax^2 + 2.25 \cdot2Bx}}{\sqrt{Ax^2 + 2Bx}} = ...
0
votes
1answer
122 views

Conditions for $v \otimes v$ to be positive semidefinite for complex $v$

I have a complex-symmetric matrix (in the sense $A=A^{T}$ not $A=A^{H}$), which is required to be positive semi-definite in the following sense (sometimes referred to as positive real): $ \Re(x^{*} A ...
2
votes
0answers
33 views

Holzer reduction of solutions of quadratic ternary forms

Suppose $(x_{0}, y_{0}, z_{0})$ is a solution to the equation $ax^2 + by^2 + cz^2 = 0$. The solution is said to be Holzer reduced if $x_{0} < \sqrt{|bc|}$, $y_{0} < \sqrt{|ac|}$ and $z_{0} < ...
1
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2answers
72 views

The no. of values of k for which $(16x^2+12x+39) + k(9x^2 -2x +11)$ is perfect square is:

I wanted to know, how can i determine the no. of values of k for which $(16x^2+12x+39) + k(9x^2 -2x +11)$ is a perfect square.($x \in R$) I have tried, since $x$ is real the discriminant must be ...
1
vote
2answers
281 views

what is the maximum number of roots of quadratic function with 3 variables?

Given the general quadratic form with $3$ variables $(x,y,z):ax^2 + by^2 +cz^2 + dxy + eyz +fzx + gx + hy + iz = 0$ satisfies $x^2 + y^2 + z^2 = 1$ I would like to ask what is the maximum number of ...
0
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0answers
160 views

what is the maximum number of roots of quadratic function with 3 variables?

Given the general quadratic form with 3 variables $(x,y,z)$: $ax^2 + by^2 +cz^2 + dxy + eyz +fzx + gx + hy + iz = 0$ satisfies $x^2 + y^2 + z^2 = 1$ I would like to ask what is the maximum number of ...
2
votes
1answer
75 views

Quadratic ternary forms

What is the difference between solubility, local solubility and global solubility when it comes to solving quadratic ternary normal forms, i.e a equation of the form $ax^2 + by^2 + cz^2 =0$? Thanks ...
1
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2answers
76 views

How to quickly tell that a quadratic only has a single root?

Hello good math wizards, I'm trying to figure out why the following equation has at most one root: $$f (t) = \textbf{x} \cdot \textbf{x} + \textbf{x} \cdot t\textbf{y} + t\textbf{y} \cdot \textbf{x} ...
2
votes
2answers
628 views

diagonalize quadratic form

I have this quadratic form $Q= x^2 + 4y^2 + 9z^2 + 4xy + 6xz+ 12yz$ And they ask me: for which values of $x,y$ and $z$ is $Q=0$? and I have to diagonalize also the quadratic form. I calculated ...
2
votes
5answers
517 views

show that the function $z = 2x^2 + y^2 +2xy -2x +2y +2$ is greater than $-3$

Show that the function $$z = 2x^2 + y^2 +2xy -2x +2y +2$$ is greater than $-3$ I tried to factorize but couldn't get more than $(x-1)^2 + (x+y)^2 +(y-1)^2 - (y)^2$. Is there any another way to ...
3
votes
1answer
122 views

Minimum of a Quadratic Form $\mathbf v^T \mathbf A \mathbf v$

I'm playing with some quadratic form for my research. In my setting, $\mathbf A$ is an $n\times n$ real symmetric matrix with only two types of eigenvalues: they are either $\frac{1}{n-x}$ with ...
3
votes
1answer
83 views

Why do we refer to certain self-adjoint operators as positive/positive definite as opposed to nonnegative/positive?

A self-adjoint linear operator $\tau$ is referred to as positive if is associated quadratic form $\langle\tau v,v\rangle\geq 0$ for all $v$, and is referred to as positive definite if $\langle\tau ...
3
votes
2answers
76 views

Where does the theory of quadratic forms fail in characteristic 2?

Let $V$ be a finite-dimensional vector space over a field $k$, and $Q$ a nondegenerate quadratic form on $V$. If the characteristic of $k$ is not 2, then we can change coördinates on $V$ so that ...
6
votes
2answers
107 views

Question on quadratic forms

I know a theorem which says: If a non-singular quadratic form (homogeneous polynomials of degree $2$) over a field $K$ represents zero non-trivially (i.e., there is a nontrivial solution of the ...
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0answers
61 views

Convergence in distribution of a quadratic form

If $Q_n=X_nM_nX_n=\sum_{i,j=1}^n X_i m_{nij}X_j$, $X_n=(X_1,...,X_n)$ where $X_j$ are iid random variables and $M_n=(m_{nij})$ is a symmetric matrix with extending rownumber in $n\to\infty$. Iam ...
2
votes
1answer
45 views

Quadratic form $\mathbb{R}^n$ homogeneous polynomial degree $2$

Could you help me with the following problem? My definition of a quadratic form is: it is a mapping $h: \ V \rightarrow \mathbb{R}$ such that there exists a bilinear form $\varphi: \ V \times V ...
2
votes
1answer
97 views

Matrix of quadratic form has to be symmetric?

On Wikipedia it is stated that any $n\times n$ real symmetric matrix A determines a quadratic form. But isn't $ax^2 + bxy + cxy + dy^2$, the quadratic form given by $v^T A v$ with $A=\begin{bmatrix}a ...
3
votes
1answer
114 views

Number of solutions of a positive integral quadratic form is finite?

Is there an easy way to see the following: Suppose Q is an integral quadratic form in $n$ variables that is positive definite, that is $Q(x) \geq 1$ for all $0 \neq x \in \mathbb{Z}^n$. Then the ...
3
votes
1answer
139 views

Any integer can be written as $x^2+4y^2$

If $n$ is a positive integer with $(n,8)=1$ and $-4$ is square $mod$ $n$ then $n$ can be written in this form: $n=x^2+4y^2$. I was using that there are x, y integers satisfying $x^2+4y^2=kn$ where ...
0
votes
1answer
63 views

Real part of quadratic form

Suppose $q$ is a quadratic form on $\mathbb{C}^n$: $q(x)=x^HAx$, with $H$ denoting the hermitian transpose. Since I am only interested in the real part of $q$, I am trying to determine a matrix $B$ so ...
2
votes
1answer
146 views

Diagonalising quadratic form

Given the quadratic form $$Q(x) = \alpha\alpha_1\alpha_2 + 2\alpha^2\alpha_1\alpha_3$$ on $\mathbb{R}^2$ where $x = (\alpha_1,\alpha_2,\alpha_3)$ in some basis I want to find the signature of $Q$ ...
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vote
2answers
127 views

Definitions and questions related to projective space $\mathbb{R}P^3$

I have the following questions regarding the definition of a quadric in a real projective space. What is the precise definition on a quadric of signature (1,1) in the projective space ...
7
votes
1answer
914 views

What's a BETTER way to see the Gauss's composition law for binary quadratic forms?

There is a group structure of binary quadratic forms of given discriminant $d$: Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$ The composition of two binary quadratic ...
0
votes
2answers
108 views

Difficulty in Quadratic equation and realtion with irrational roots

One root of the quadratic equation $ax^2 +bx + c=0$ is $\dfrac{2}{\sqrt{3} + \sqrt{5}}$. If $\frac{c}{a}$ is rational, then how do we find the other root. the answer given is that the other root is ...
0
votes
0answers
260 views

Writing a quadratic form as a sum of squares

Let $Q(x_1,x_2, \ldots ,x_n)$ be a positive definite real quadratic form in the variables $x_1, \ldots ,x_n$. It is not hard to see that the function $f(x_1, x_2, \ldots ,x_n)=\frac{Q(x_1,x_2, \ldots ...
0
votes
1answer
104 views

Problem on hyperbolic hyperboloid generated by a rotation

This is the problem: In $\mathbb{E}^3$ we consider the conic $\gamma$ of equations $x=yz-2=0$ , the line $a$ of equations $x=y+z=0$ and the surface $Q$, that is generated by the rotation of $\gamma$ ...
1
vote
1answer
23 views

solving for one variable in terms of others

A question from Steward's Precalculus textbook 5th, Pg 55, the original formula is $$h=\frac{1}{2}gt^2+V_0t$$ the question asks to write the formula in terms of $t$, the answer is ...