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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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 ...
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2answers
354 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 ...
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0answers
49 views

low rank decomposition of composite PSD matrix

The matrix $M=AVA^T - BCB^T +D$ is known to be positive semidefinite (PSD), where $V, C, D$ are each diagonal matrices with positive values, and $V, C$ has small size when compared to the size of $M$. ...
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1answer
64 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}} = ...
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1answer
106 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 ...
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0answers
32 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} < ...
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70 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 ...
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2answers
260 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 ...
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0answers
159 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 ...
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1answer
74 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 ...
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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} ...
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2answers
586 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 ...
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5answers
501 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 ...
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1answer
118 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 ...
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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 ...
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2answers
74 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 ...
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2answers
104 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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60 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 ...
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1answer
42 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 ...
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1answer
92 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 ...
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1answer
110 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 ...
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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 ...
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1answer
61 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 ...
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1answer
143 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
124 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
487 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 ...
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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 ...
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0answers
256 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 ...
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1answer
92 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 $\gamma$ ...
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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 ...
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117 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 ...
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1answer
87 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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106 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 ...
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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.
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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 ...
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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 ...
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241 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 ...
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1answer
155 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 ...
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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 ...
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0answers
30 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 ...
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5answers
133 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
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1answer
206 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? ...
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68 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 ...
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1answer
304 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 ...
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1answer
642 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 \\ ...
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1answer
76 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 ...
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1answer
125 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. ...
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2answers
143 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 $x=\cosh t,y=\sinh t$. ...
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1answer
56 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 ...