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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0
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1answer
145 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
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0answers
34 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
74 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
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2answers
346 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
167 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
76 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
659 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
533 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
126 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
84 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
77 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
110 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 ...
1
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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
49 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
102 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
120 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
143 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
66 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
149 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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2answers
132 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 ...
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1answer
961 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
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2answers
109 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
270 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
116 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
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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 ...
3
votes
2answers
130 views

Quadratic Equation with “0” coefficients

Let's say I have two objects $x$ and $y$ whose position at time $t$ is given by: $$ x = a_xt^2+b_xt+c_x \\ y= a_yt^2+b_yt+c_y $$ And I want to find which (if any) values of $t$ cause $x$ to equal ...
3
votes
1answer
91 views

Question about the definition of representability of a quadratic form

Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find ...
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3answers
129 views

$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form

Prove that every $n$-linear alternating form on a vector space of dimension less than $n$ is the zero form.
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1answer
50 views

$n$-linear form: An Interpretation

What is a good example of an $n$-linear form that is more familiar to a student learning at an elementary level? EDIT: I'm just trying to show that every $n$-linear alternating form on a vector ...
1
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1answer
45 views

How to show that $A=B-C$

How to show that for a real symmetric matrix $A,~A$ can be written as $A=B-C$ where $B,C$ are positive definite real symmetric matrices? Please help me ! I'm clueless.
1
vote
1answer
71 views

Solving quadratic form $\mathbf{x}^\mathrm{T}\mathbf{A}\mathbf{x} = c$ for $\mathbf{x}$

This is a simple question I hope, is there an easy way to solve: $$ \mathbf{x}^\mathrm{T}\mathbf{A}\mathbf{x} = c $$ for $\mathbf{x}$? (Assume $\mathbf{A}$ is positive definite). Geometrically the ...
1
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1answer
24 views

Eigenvalues of $\sum_{i=1}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}$

Consider the cuadratic form $$ \mathbf{x}^{\intercal}Q\mathbf{x} = \frac{x_1^2}{\lambda_1} + \sum_{i=2}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}\ . $$ Is it true that the eigenvalues of $Q$ are ...
6
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2answers
314 views

Coercive bilinear form on Hilbert space

I need to show the two following results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. Consider a continuous symmetric bilinear form $B$ on a ...
1
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1answer
167 views

prove that determinant is a quadratic form

let $V$ be a vector space of all $2 \times 2$ hermitian matrices with entries from $\mathbb C$, over the field $\mathbb R$. prove that $q(v)=\det(v)$ is a quadratic form. I tried to prove that ...
0
votes
1answer
30 views

Quadratic fit check

I've performed LS fit to data in order to fit the following quadratic function: $$f(x,y) = A~x^2 + B~y^2 + C~x~y + D~x+E~y +F$$ Now, I would like to check that the fitted function looks like a ...
1
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0answers
32 views

Solve I.V.P for differential using quadratic form

Solve the i.v.p for $y''+4y'+5y=0, y(\frac{\pi}{2})=1/2, y'(\frac{\pi}{2})=-2$ I solved using the quadratic form. and I got $\lambda = \frac{(-4 \pm 2i)}{2}$, which for $\lambda 1,2= 2+2i$. And then ...
1
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5answers
138 views

How do you determine whether the quadratic form is positive and negative definite?

How do you determine whether the quadratic form $Q(x,y) = 2x^2 - 4xy + 5y^2$ is positive definite, negative definite, or indefinite? Could someone show step by step with explanations? Thank you
6
votes
1answer
207 views

How to prove that $ E:=ABC D $ is also positive definite?

Now I think this is true: Let $A$, $B$, $C$ and $D$ be symmetric, positive definite matrices and suppose that $E:=ABCD $ is symmetric. How might I prove that $E$ is also positive definite? ...
0
votes
1answer
81 views

Generating vectors of the face-centered cubic lattice

I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by ...
9
votes
2answers
510 views

Generalizing the 290 theorem.

I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic ...
1
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1answer
821 views

Real and complex canonical forms of quadratic form

How do I find the canonical form of $$q_1(x,y,z)= 4x^2 +4xz+2yz$$ Now I have put it in matrix form as: $$\left( \begin{matrix} 4 & 0 & 2 \\ 0 & 0 & 1 \\ ...
1
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1answer
87 views

Quadratic Forms in Non-Linear Optimization

This is a rather trivial question but I am having a great deal of trouble: Let $f(x) = (1/2)xQx-xb$ and $E(x) = (1/2)(x-x^*)Q(x-x^*)$ then $E(x) = f(x) + (1/2)x^*Qx^*$ where $x,x^*,b$ are vectors ...
2
votes
1answer
135 views

In quadratic form, how would symmetric matrix $A$ would change under coordinate change?

In http://en.wikipedia.org/wiki/Quadratic_form, Let $q$ be a quadratic form defined on an n-dimensional real vector space. Let $A$ be the matrix of the quadratic form $q$ in a given basis. ...
2
votes
2answers
157 views

Finding a parametrization of a hyperbola who has a fixed signature

How do I find parametrization of the hyperbola $x^2-y^2=1$ which is the unit sphere of a quadratic form with signature $(1,-1)?$ The only parametrization that comes to mind is $x=\cosh t,y=\sinh t$. ...
1
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1answer
69 views

Determining a norm from a quadratic form

If $B$ is a quadratic form over some space $V$, what is the norm determined by $B$? Is this the inner product $\langle Bu,Bv\rangle$? If not, and it is not possible to determine a norm from knowing ...
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2answers
139 views

Lucky Lattice Points

How many lattice points lie on the sphere given by following equation ? $$x^2+y^2+z^2=2013$$ Hint: A lattice point has integer coordinates.
2
votes
0answers
32 views

$U$ is isotropic $\implies$ $U\subset U^0$

Let $(V,Q)$ be a quadratic module and $U$ is a subspace of $V$. Serre (A Course in Arithmetic, p. 29) claimed that the following is evident: $U$ is isotropic $\iff$ $U\subset U^0$. In other ...
3
votes
6answers
379 views

Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
0
votes
1answer
77 views

Rewriting a quadratic Matrix equation as a quadratic vector equation

Consider the set of $N \times N$ matrices $\{W_i\}_{i=1}^{i=L}$, set of $N \times 1$vectors $\{g_i\}_{i=1}^{i=L}$ and $\{h_i\}_{i=1}^{i=L}$. Now consider the following sum \begin{align} ...