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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9
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0answers
21 views

Intersection of two hyperplanes

$G$ and $H$ are hyperplanes in $\mathbb{P}_n$ with coordinates $g=(g_0, \ldots, g_n)$, $h=(h_0, \ldots, h_n)$. How can I find a symmetric matrix $A_Q$ of a quadric $Q$ with $ Q = G \cap H$, where ...
1
vote
0answers
14 views

What is the meaning of projectivized quadratic forms?

I have a quadratic form $$C_1=z_1^2 + z_2^2 +z_3^2.$$ What does it mean to projectivize $C_1$? I am guessing that it is $$[C_1]=\{C : C=\lambda C_1, \ \lambda \in \mathbb{C} \}.$$ Is this correct? ...
1
vote
0answers
32 views

Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$

Kindly see the question at the end of post. Solutions to the system of three equations, $$\begin{aligned} a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\ a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\ abcd\, ...
2
votes
1answer
50 views

Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

Sinha’s theorem can be stated as, excluding the trivial case $c = 0$, if, $$(a+3c)^k + (b+3c)^k + (a+b-2c)^k = (c+d)^k + (c+e)^k + (-2c+d+e)^k\tag{1} $$ for $\color{blue}{\text{both}}$ $k = 2,4$ ...
1
vote
1answer
21 views

Is the sums of sqaures (without zero) a multiplicative group?

I'm taking a course in quadratic forms at the moment. There is no text book or notes so all I have to go from is what the teacher writes on the blackboard. We had the following lemma some weeks ago: $ ...
4
votes
3answers
82 views

For what $a,b$ such that $ax^2+by^2 = z^2$?

This post made me think about this question. What is the criterion on positive integer $a,b$ such that, $$ax^2+by^2 = z^2$$ can be solved in positive integers $x,y,z$? (Three broad classes are: 1) ...
0
votes
1answer
18 views

Show that $x \mapsto \left( x^{\top} \sigma x , -\mu^{\top}x \right)^{\top}$ transforms a given set into a convex set.

Let's say you have a covariance matrix $\sigma$ and a vector of expected returns $\mu$. Basically $\sigma$ is a symmetric matrix with positive eigenvalues and we can safely assume that $\mu$ is just a ...
0
votes
1answer
92 views

Classify the surface: $x^2+y^2-z^2+2xy-2xz-2yz-y=0$

I need to determine what shape the surface is and justify it. Now wolfram alpha tells me that this particular surface is a hyperbolic paraboloid which has the general form: $\alpha x^2-\beta ...
0
votes
1answer
12 views

Is there a way to analytically find a stationary point along an arbitrary line in a multivariable quadratic function?

Let's say I'm working with a quadratic function with an equation of $f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^TA\mathbf{x} - b^T\mathbf{x}$. Now, let's take a direction $\mathbf{p}$ and transform the ...
0
votes
1answer
19 views

Composition of binary quadratic forms as matrix operations

It is easy to see that any binary quadratic form $a^2 + 2bxy + cy^2$ is the same as $XAX^T$ where $X = [x, y]$ and $A = \begin{bmatrix}a & b\\b & c\end{bmatrix}.$ The composition of two ...
1
vote
1answer
22 views

Show that quadratic form $z_0^2+z_1^2+z_2^2$ is invariant under $SO_3(\mathbb{R})$

Let $z=(z_0,z_1,z_2)$. We thus have $z_0^2+z_1^2+z_2^2 = z^Tz=||z||$. Showing invariant means is this what I need to show: $$\forall A\in SO_3(\mathbb{R}), \ ||Az||=||z||?$$ But this is clear from ...
0
votes
0answers
8 views

Solution to a particular quadratic vector equation

I have to solve for $y$ from a quadratic vector equation of the form $$y^{T}Py + 2q^{T}y = 0,$$ where $P \in \mathbf{R}^{n \times n}$ is positive semidefinite, and $q \in \mathbf{R}^{n}$. I got some ...
0
votes
2answers
51 views

Minimum value of a positive definite binary quadratic form along integers

Is there a formula for the least non-zero value of $$f(x,y):=ax^2+bxy+cy^2$$ as $x,y$ assume integer values? Here $a,b,c$ are integers with $a,d>0$ and $b^2-4ac<0$.
0
votes
1answer
32 views

Linear Algebra quadratic forms diagonalization

I have a question that reads: Diagonalize the quadratic form $A(x,x) = 2x^2 - 1/2 y^2 -2xy - 4xz$ by completing the squares, and find the change of basis matrix and the new basis in which A will be ...
0
votes
1answer
29 views

Linear Algebra Quadratic Form Diagonalization

I asked this question the other day but I still didn't understand it. Hopefully someone can get through to me this time. I have a question that reads: Diagonalize the quadratic form A(x,y) = 3x^2 - ...
0
votes
0answers
24 views

Addition and Subtraction of Quadratic Forms

Working through matrix algebra, I have to solve the following expression, which is composed of quadratic forms. My question is how do you simplify this expression? Please be as detailed as possible. ...
5
votes
1answer
119 views

An argument from a blog article of Terence Tao

Let $A_1, A_2, A_3, \ldots , A_m$ be positive semi-definite Hermitian matrices and then consider the polynomial $p(z,z_1,z_2,\ldots,z_m) = \det(z+z_1A_1 + z_2A_2 + \cdots+z_mA_m)$ Now Tao argues that ...
0
votes
2answers
41 views

What is the polarization identity?

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and ...
3
votes
2answers
45 views

Diagonalizing Quadratic Forms. Linear Algebra

I have a question that reads: Diagonalize the quadratic form $A(x,y) = 3x^2 -12xy + 7y^2$ by completing the square. What is diagonalization? Is that when I should find the eigenvector matrix, ...
0
votes
0answers
36 views

Quadratic form and basis

I computed (1). The matrix A can be diagonalized as P^T A P=D where $$D=\left(\begin{array} -6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 12 \end{array} \right)$$ ...
3
votes
2answers
79 views

An annoying Pell-like equation related to a binary quadratic form problem

Let $A,B,C,D$ be integers such that $AD-BC= 1 $ and $ A+D = -1 $. Show by elementary means that the Diophantine equation $$\bigl[2Bx + (D-A) y\bigr] ^ 2 + 3y^2 = 4|B|$$ has an integer ...
0
votes
1answer
40 views

Finding a matrix $P$ such that $P^TAP$ is diagonal

I'm kind of confused on some linear algebra. On a previous question I was given a quadratic form and I had to find a matrix $P$ such that $P^TAP$ is diagonal. I did this by using a suitable change of ...
0
votes
0answers
22 views

determinant of the covariance matrix of a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol ...
1
vote
3answers
45 views

How would one solve this system

$$12x^2=6z\\2y=-z\\6x-y=7$$ It's been many years since I've dealt with system equations, and now find myself in need to solve them. I am not quite sure what to do; I am interested in finding $x$ and ...
0
votes
2answers
39 views

Equivalent quadratic form with 4 varibles

Consider two quadratic forms: $Q(x,y,z,w)=x^{2}+y^{2}+z^{2}+bw^{2}$ and $P(x,y,z,w)=x^{2}+y^{2}+czw$. For what type of values of $b$ & $c$ (real or complex or negative or positive or zero) $P$ ...
0
votes
1answer
29 views

If $x\in \mathbb{R}^n$ and is a unit vector, why is $\sum\limits_{j,k=1}^n |x_j||x_k| < n^2$?

This is an excerpt of a larger proof: Other pertinent information: $A$ is a positive definite $n \times n$ matrix The set $C$ is the unit sphere I don't get the last inequality: $\gamma \sum ...
0
votes
1answer
24 views

Help explain the geometry being described in this paragraph

In A First Course in Optimization Theory by Sundaram I read on page 51: Given a quadratic form $A$ and any $t\in\mathbb R$, we have $(tx)'A(tx)=t^2x'Ax$, so the quadratic form has the same sign ...
1
vote
0answers
21 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
0
votes
1answer
23 views

Signature of a quadratic form

Let $P$ be a $n$_degree polynomial over $\mathbb{R}$. Let the quadratic form on the vector space of $n$_degree polynomials: $$H(P)=\int_0^\infty e^{-x^2}P(x)P(-x)dx$$ What is the signature of $H$? ...
1
vote
1answer
35 views

Find $a^2 + b^2+c^2$

Given $a^2+2b = 7$ $b^2+4c = -7$ $c^2+6a = -14$ Find $a^2 + b^2 + c^2$ The answer was an Integer I tried to solve it by making $a$ the subject of the equation and substituting in others but ...
5
votes
2answers
78 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
0
votes
1answer
31 views

A question on a nonnegative quadratic form

Denote $x,y,z$ as variables, and $a,b,c$ as coefficients. Suppose $a\leq b\leq 0\leq c$ and $a+b+c=0$. Could anyone help me prove whether the following quadratic form positive semi-definite? ...
1
vote
1answer
51 views

Non-standard quadratic matrix equation

I have an equation that looks like the following: $$ A\cdot\mathrm{diag}(x)\cdot x + B\cdot x + c = 0 $$ where $A, B, C \in \mathbb{R}^{n \times n}$ and $x, c \in \mathbb{R}^n$. $ x $ is unknown. ...
1
vote
1answer
62 views

A problem on positive semi-definite quadratic forms/matrices

Suppose $a+b+c=0$ and (without loss of generality) $a\leq b\leq 0\leq c$, $a^2+b^2+c^2=1$, is the following quadratic form positive semi-definite? Thank you very much. \begin{equation*} \begin{split} ...
1
vote
1answer
18 views

Scaling variables in homogeneous equation of degree two in a,b,c

The problem I'm having trouble with is: Let $a,b,c$ be nonzero real numbers and let $a^2 - b^2 = bc$ and $b^2 - c^2 = ca$. Prove that $a^2 - c^2 = ab$. The solution strategy given in the course was ...
0
votes
1answer
47 views

Equivalence for binary quadratic forms with positive square discriminant

I recently encountered an interesting proposition without proof: If $f(x,y)$ is a quadratic form whose discriminant is a non-zero perfect square, then $f(x,y)$ is equivalent to a form $a*x^{2} + ...
3
votes
0answers
36 views

Freeman Dyson's identity for the modular discriminant $\Delta$

In his 'Introduction to Modular Forms', Don Zagier states the Freeman Dyson's identity : $$\Delta(\tau)=\sum_{\substack{ (x_1,\ldots,x_5)\in \mathbb{Z}^5 \\ x_1+\cdots+x_5=0 \\ x_i \equiv i ...
0
votes
1answer
27 views

Quadratic equation form

I have the relation $u=\sqrt{(a_1+b_1t)^2+(a_2+b_2t)^2+(a_3+b_3t)^2} \tag 1$ I need to write $t$ as a function of $u$ ($t=f(u)$). How will I get that ? NB: $a_1,a_2,a_3,b_1,b_2,b_3$ are ...
0
votes
0answers
18 views

Reduce to diagonal form.

Problem is to reduce $5X^2+3Z^2+4XY-4YZ+6ZX$ into diagonal form over $\mathbb{R}$. With my knowledge, We ned to make a non-singular variable transformation so that above form comes into a form like ...
0
votes
0answers
18 views

Finding discriminant of this quadratic form

My question is what are the discriminants of $X^2+Y^2 $ and $X^2-Y^2$ over $\mathbb{R}$ and $\mathbb{C}$ and why? It should be $1$ and $-1$ respectively over $\mathbb{R}$. But shouldn't they be same ...
0
votes
0answers
28 views

Proving $\dim(W)+\dim(W^\perp)=\dim(V)$

I have to prove if $V$ is non-degenerate, then for any subspace $W$ of $V$, $1)$ $\dim(W)+\dim(W^\perp)=\dim(V)$, $2)$ $W^{\perp\perp}=W$ $3)$ $\operatorname{rad}(W)=W\cap W^\perp$ I was doing ...
1
vote
1answer
36 views

Discriminant of a Quadratic form

Let $V$ be a vector space over field $K$ and $Q$ is the quadratic form on it, and $A$ be the matrix w.r.t. $e_1,e_2,...e_n$ of $V$. Now $discr(Q)$ is defined as $det(A)$ mod ${K^{*}}^{2}$. Now my ...
2
votes
1answer
66 views

How to show that two quadratic forms are equivalent?

To show to quadratic forms are not equivalent, we can find rank, or discriminant or some element which is represented by either one only etc. But Is there a general criterion to show that two ...
0
votes
2answers
28 views

Solve a system of equations when one is linear and the other is quadratic

$x+y=3m$ $xy=2m^2$, $m$ is the parameter. I came to this $2m^2-3mx+xy=0$. The solutions have to be:$(m,2m),(2m,m)$. But I can't understand what is the role of this parameters, I don't know how to ...
0
votes
0answers
14 views

Prove that every reproducing kernel is a positive matrix (and vice versa)

Let $\mathcal{H}$ be a functional hilbert space (defined over a set $S$) with a reproducing kernel K. Prove that: a) $K$ is a positive matrix means the queadtric form is positive, i.e ...
1
vote
3answers
33 views

System of two equations with two unknowns - can't get rid of $xy$

The system is: $x^2 + 2y^2 + 3xy = 12$ $y^2 - 3y = 4$ I try to turn $x^2 + 2y^2 + 3xy$ into $(x + y)^2 + y^2 + xy$ , but it's a dead end from here. Can anyone please help?
0
votes
2answers
93 views

Solving homogeneous quaternary quadratic Diophantine equation

Given the equation $w^2+x^2+y^2+z^2=wx+wy+wz+xy+xz+yz$, how does one systematically enumerate all non-negative integer solutions $\{w,x,y,z\}$?
0
votes
0answers
20 views

Is $f(\alpha,\beta) = (x_{1}-y_{1})^2 + x_2 y_2 $ a billinear form?

Where $f: \Bbb R^2 \times \Bbb R^2 \to \Bbb R, \alpha = (x_1,x_2), \beta = (y_1,y_2)$ Actually how to determine the matrix entries for the coefficient for $x_1^2, y_1^2.$ Is there any method without ...
0
votes
0answers
31 views

Index of inertia for quadratic forms

How to find the indies of inertia and canonical quadratic form of: $ f= 2x_2^2 -x_3^2 + 2x_1x_2 - 4x_1x_3 $ so I guess our matrix would look like: $\begin{pmatrix} 0 & 1 & -2 \\ 1 & 2 ...
0
votes
2answers
41 views

Prove a quadratic form is positive definite

I want to prove - without using eigenvalues- that the quadratic form $$q(x,y)=Ax^2+2Bxy+Cy^2$$ is positive definite iff $A>0$ and $AC-B^2>0$ This exercise was taken from a practice for a ...