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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47 views

Quadratic Equations GRE Quants

It would be very useful if someone can give me an answer to this question with a proper explanation. One of the factors of the equation $x^2 +9x + c$ is $(x+11)$, where $c$ is a constant. Which of ...
3
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0answers
152 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...
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0answers
49 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\, ...
3
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3answers
274 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 $\color{blue}{\text{both}}$ $k = 2,4$ ...
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1answer
29 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
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3answers
129 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) ...
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1answer
26 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
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1answer
150 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: $\alpha x^2-\beta y^2-z=0$...
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1answer
38 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 ...
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2answers
58 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 ...
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1answer
31 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
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2answers
172 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 $b^2-4ac<0$.
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1answer
70 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 ...
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1answer
67 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 - ...
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1answer
158 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 ...
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2answers
72 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
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2answers
175 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, ...
3
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2answers
114 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 ...
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1answer
123 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 ...
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3answers
59 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 $...
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2answers
102 views

Test whether the foloowing quadratic forms are Equivalent

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) $P$ ...
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1answer
31 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 \...
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1answer
42 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 ...
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0answers
42 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
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1answer
89 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 $H$? ...
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1answer
39 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 ...
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2answers
102 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
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1answer
48 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? \begin{...
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1answer
77 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. ...
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1answer
150 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} ...
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1answer
32 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 ...
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1answer
215 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
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0answers
87 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
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1answer
37 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 ...
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1answer
63 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 ...
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1answer
792 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 binary(...
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2answers
37 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 ...
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3answers
36 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?
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2answers
233 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 $\{w,x,y,z\}$?
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2answers
305 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 ...
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1answer
55 views

Inequality of expectation of a quadratic form

I was reading a proof in a paper. Let $X$ and $Y$ be two possibly correlated $K$-dimensional random vectors. Suppose $\mathrm{E}(YY^T)=I$, where $I$ is an $K\times K$ identity matrix. By Cauchy-...
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1answer
39 views

Terminology: what is the “generic character” of a ternary quadratic form?

The title says it all: What is the "generic character" of a ternary quadratic form? Motivation: I'm reading a really old paper, and the author refers to this terminology without any further ...
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3answers
59 views

A simple system of equations

I'm trying to refresh my school math knowlegde and have trouble solving a simple system of equations: $\begin{cases} x + xy + y = -3,\\ x - xy + y = 1. \end{cases}$ I derive $y$ from the second: $y ...
0
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1answer
32 views

Implications of a prime square dividing a binary quadratic form

Let $u,v$ be positive integers with $\gcd(u,v)=1$, let $k\ge 3$ be an odd integer, and fix a prime $p$. Now what are the implications of $p^2 \mid (u^2+kv^2)$? I know implications in certain cases, ...
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2answers
119 views

how many solutions to non-linear simultaneous equations

I'm doing a Lagrange multiplier optimization problem, and I wound up with the following simultaneous equations: $2x + 1 -2\lambda x = 0$ $4y-2 \lambda y = 0$ $6z-2 \lambda z = 0$ $-x^2 - y^2 - z^2 + ...
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0answers
40 views

can someone break this quad formula down for me?

Can someone explain how this person yield the stuff on the right side using quad formula?
4
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1answer
51 views

Same quadratic forms on $\mathbb R^n$

Let $q$ be an inner product on $\mathbb R^n$ and $Q$ be its matrix expressed in the canonical basis of $\mathbb R^n$. Assume that the group $$SO(q)=\{A\in M_n(\mathbb R) \ | \ A^TQA=Q\}$$ of ...
0
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1answer
198 views

Discriminant of a ternary quadratic form

What is the discriminant of a ternary quadratic form $x^2-y^2+z^2-2xy+4yz-6xz$? The answer says, first make it $a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{23}yz+2a_{13}xz$, and then the discriminant ...
0
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1answer
95 views

Quadratic forms — rank of matrix

Assume that $M$ is the matrix of some quadratic form (over any field of characteristic not $2$) and set $$Q(\overline{x})=\overline{x}^tM\overline{x}$$ We can replace $M$ by the symmetric matrix $N=\...
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1answer
49 views

Polynomial and super-symmetric tensor

A quadratic function uniquely determines a symmetric matrix. Ok that’s easy. Now a homogeneous polynomial function $f(x)$ also uniquely determines a super-symmetric tensor. My question is how do I ...