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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43 views

How to solve the quadratic form

I am a physicist and I have a problem solving this \begin{equation} Q(x)=\frac{1}{2}(x,Ax)+(b,x)+c \end{equation} In a book it says that: "The minimum of Q lies at $\bar{x}=-A^{-1}b$ and ...
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2answers
44 views

Fast verification of solution to x'Ax<C

Assume we have some complex vector with N dimensions $\vec x$. We need to verify if this is a valid solution to: $\vec x^HA\vec x<C$ where $A$ is a Hermitian matrix and $C$ is some real ...
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0answers
28 views

Can systems of equations of diagonal quadratic forms be solved by Gaussian Elimination

Can the following system of equations be solved using Gaussian Elimination? $$ \begin{bmatrix} s_{00} & s_{01} & s_{02} & s_{03}\\ s_{10} & s_{11} & ...
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0answers
45 views

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

I am working on a problem in which I must classify the surface described by the following equation $$x^2 + y^2 - z^2 + 2xy - 2xz - 2yz - y = 0.$$ I have looked at this Stack Exchange discussion (on ...
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2answers
100 views

Given $p \equiv q \equiv 1 \pmod 4$, $\left(\frac{p}{q}\right) = 1$, is $N(\eta) = 1$ possible?

Given distinct primes $p$ and $q$, both congruent to $1 \pmod 4$, such that $$\left(\frac{p}{q}\right) = 1$$ and obviously also $$\left(\frac{q}{p}\right) = 1$$ is it possible for the fundamental unit ...
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3answers
148 views

Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution

Help solving over the integers: $$ax^2+by^2+cz^2+dxy+exz+fzy=0$$ where $(x_0,y_0,z_0)$ is a known integral solution and $a,b,c,d,e,f$ are integral coefficients. I found in Tito Piezas' identities the ...
3
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2answers
62 views

Equivalence of Quadratic Forms that represent the same values

An integer quadratic form is a function $Q(x,y) = ax^2 + bxy + cy^2$ where the numbers $a,b,c \in \mathbb Z$. Call the set of values a quadratic forms takes on $V(Q) = \{ Q(x,y) \in \mathbb Z | x,y ...
2
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1answer
28 views

Non-trivial kernel if quadratic form is indefinite

I am wondering the following: if $f:\;V\to \mathbb{R}$ is a quadratic form that is neither positive or negative definite, must its kernel be non-trivial? Here quadratic form means $f(v)=g(v,v)$ where ...
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0answers
30 views

Defining a Euclidean Structure on a real vector space.

This comes from a homework question: For $\bf x, \bf y$ $\in \mathbb{R}^n$, put $\langle {\bf x}, {\bf x} \rangle$ = $\sum_{i=1}^n 2x_i^2 - 2\sum_{i=1}^{n-1}x_ix_{i+1}$. Show that the corresponding ...
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1answer
35 views

Question about the proof of simultaneous diagonalization of quadratic forms

I'm trying to understand a proof about this theorem and I find myself stuck in a step. Let's consider two quadratic forms $\langle q, Aq \rangle$ and $\langle q, Bq \rangle$, where the first one is ...
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2answers
43 views

Show $p$ prime s.t. $p \not\equiv 1 \mod 3$ is represented by the binary quadratic equation.

I am working on the following question: Let $p>3$ be a prime such that $p \not\equiv 1 \mod 3$. Show that $p$ is not represented by the binary quadratic equation $f(x, y) = x^2 + xy + y^2$. I ...
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0answers
9 views

Connection between maximizing a quadratic form and maximal variance

In order to find the "directions of maximum variance" of $X$ one finds the eigen decomposition of the variance covariance matrix $X^tX$. I have seen the eigenvector problem cast as maximizing the ...
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0answers
10 views

Is this the correct way to interpret results from the principal minor test?

Am I correct in thinking that, given all the principal minors $A_{i}$ of the matrix associated with a quadratic form $Q(x,y)$, the Principal Minor Test states that $Q(x,y)$ is Positive Definite if ...
2
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1answer
44 views

Prove a binary quadratic equation has specific number of solutions

How do I show that the binary quadratic equation $f(x, y) = x^2 + xy + y^2 = 1$ has exactly $6$ solutions? The discriminant is $-3$, so I cannot use Pell's Equation ($x^2 - dy^2 = p$, where $d>0$ ...
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1answer
26 views

When is $SO(m,n)$ simple as a Lie group? What are the Zariski and Euclidean components?

Let $SO(m,n)=\operatorname{SO}(m,n)(\mathbb{R})$ denote the real $(m+n) \times (m+n)$ matrices, with determinant $1$, which preserve the quadratic form $x_1 + \cdots + x_m - x_{m+1} \cdots - x_{m+n}$ ...
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1answer
19 views

Quadratic form and spectral theorems

Let P be a quadratic form with real coefficients in $\mathbb{R}^{n}$ such that $P^{-1}(1)$ is non-empty and compact (bounded). Show that there is an orthogonal transformation that maps $P^{-1}(1)$ ...
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1answer
23 views

Is $x^TAy$ convex or concave, for $x,y$ not identically equivalent?

It is well known if we had something like: $f(x) = x^TQx$ A quadratic form, is positive semidefinite of $Q$ is positive semidefinite How is the structure of $f(x,y) = x^TQy$ analyzed? i.e. what ...
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2answers
60 views

Diagonalisation of a quadratic form.

Find a coordinate transformation diagonalizing the quadratic form. Interesting in answering number 2. So, here is my approach:- Step 1:- Write the matrix representation of the equation, that is A= ...
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2answers
44 views

solution of quadratic equation n unknown

My question is as follows: Let the equation $V^{\top}MV=F$. Such as $V^{\top}=(x_1,x_2,...,x_n)$ a line vector of n unknown coefficients, M a known diagonal matrix (of size n) and F a real number ...
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1answer
45 views

Definition of equivalence of quadratic forms.

I am reading A Course in Arithmetic by J-P Serre. Definition $7$ on page $32$ says Two quadratic forms $f$ and $f'$ are called equivalent if the corresponding modules are isomophic. I am not ...
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3answers
115 views

Writing an expression as a sum of squares

I'd like to write $2xy+2xz+2yz$ in the form $a(\cdots)^2+b(\cdots)^2+c(\cdots)^2$ where each blank space is a linear combination of $x,y,z$. The closest I have is: ...
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1answer
51 views

Which integer from 1 to 20 does the quadratic form <7,11> over Q represent? [closed]

Which integer from 1 to 20 does the quadratic form $<7,11>=7x^2+11y^2$ over $\mathbb{Q}$ represent? This is an exercise from chapter 1 of Lam's book, Introduction to Quadratic forms over fields. ...
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0answers
14 views

Find curves that generate a dual basis to a space of one-forms

Let $V$ be the vector space of one-forms on the plane that have quadratic functions as coefficients of $dx$ and $dy$, with basis $\{x^2dx,xy\;dx,y^2dx,x^2dy,xy\;dy,y^2dy\}$. For any curve $\Gamma$ ...
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1answer
34 views

Quadratic Form - find a minimal scalar $m \in \Bbb R$ such that $q(x,y,z) \le m(x^2+y^2+z^2)$

Let $q (x,y,z)$ be a quadratic form, $$q(x,y,z)=2zx+4yz-2xy $$ $$V=\Bbb R^3$$ Find a minimal scalar $m \in \Bbb R$ such that $$q(x,y,z) \le m(x^2+y^2+z^2)$$ for all $x,y,z \in \Bbb R$. ...
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0answers
20 views

Quadratic form of Kroenecker products of skew-symmetric matrices

I am trying to understand under which conditions on $P=P^\top>0$ , $C=C^\top$, the following quadratic form is zero: $$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$ ...
3
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2answers
59 views

Bilinear maps and functions of the form $(x,y) \mapsto ux^2 + 2vxy + wy^2$

I was recently reading from the book "Vectors, Pure and Applied", $\S 16.1$ on bilinear forms. It begins In section 8.3 we discussed functions of the form $$(x,y) \mapsto ux^2 + 2vxy + wy^2$$ ...
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1answer
60 views

Homework question on quadratic forms and change of coordinates

I have been given the following question (on a homework): i) Write down the symmetric matrix A corresponding to the quadratic form $q(v)= wz-xy$ in the 4 variables $w,x,y,z$. I have the matrix A = ...
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3answers
63 views

Express a quadratic form in three variable in the format $x^tAx$ using a substitution $x=Py$

I was asked to determine if a quadratic form is positive definite. To do so I must convert in the format $x^tAx$ using a substitution x=Py. So that "it can be written in diagonal form". $$Q(x,y,z) = ...
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3answers
69 views

Finding the matrix of a quadratic form

I want to find the matrix of quadratic form $Q= \sum^p_{i=1} (y_i - \bar y)^2$. Please help me finding it. For example I have found the quadratic form matrix for $Q= p\bar y^2$ as follows: ...
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1answer
33 views

Compute the dimension of the space of quadratic forms

We were asked the following: "Compute the dimension of the space of quadratic forms on $V=\mathbb{R^2}.$ Compute also the dimension of the space of symetric forms on $\mathbb{R^2}$, ...
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0answers
83 views

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

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
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0answers
25 views

Quartic polynomial in ten variables

I have a quartic form, i.e. a homogeneous 4-th degree polynomial, in ten real variables and the inequality: $f(x_1,\ldots, x_{10}) \geq c$, for some $c>0$, which I believe that geometrically ...
3
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0answers
16 views

$q_I$ primitive as a quadratic form? [closed]

Let $I \subset \mathcal{O}_k$ be an ideal, $N(I) = [\mathcal{O}_K : I] = |\mathcal{O}_K/I|$. Define $q_I$ be $q_I(x) = N_{K/\mathbb{Q}}(x)/N(I)$. Is $q_I$ primitive as a quadratic form?
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20 views

Optimizing the sum of powers of positive quadratic functions

In my research I have come across the following optimization problem. \begin{equation} \begin{array}{c} maximize \hspace{1cm} \sum \limits_{n=1}^{N}\left(\mathbf{x}^{T}A_n \mathbf{x}\right)^{k}\\ ...
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1answer
25 views

Matrix of quadratic form (in Serre's general notion)?

I am currently reading Serre's book on arithmetic. In chapter four (page 27) he defines a general notion of the quadratic form as: Let $V$ be a module of a commutative ring $A$. A function $Q$ is ...
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1answer
29 views

Solution of quadratic diophantine equations

Is there any algorithm so that solution to the following equation can be found? $(x+a)^2-y^2=c$ where $c$ and $a$ is a constant. It is similar to Pells eqution with a variation where $D=1$. I am new ...
0
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1answer
33 views

Is the following inequality correct?

I'm trying to understand whether the following inequality is correct. Let $Y,X$ be random variables and $f(X)$, $n\times 1$-dimensional function of $X$. It is claimed that $$\begin{aligned} & ...
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32 views

Order of the sum of elements of the inverse of a matrix

For each $T$, let $A_T$ be a $T\times T$ matrix of real numbers. let $e_T$ be the $T\times 1$ vector of ones. Assume that the sum of all entries of the matrix $A_T$ divided by $T^2$ is limited as $T$ ...
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71 views

Vieta Jumping and Hurwitz 1907

Today I proved finiteness for the problem here: Is it true that $f(x,y)=\dfrac{x^2+y^2}{xy-t}$ has only finitely many distinct integer values with $x,y$ positive integers? namely: IF we have ...
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0answers
14 views

Differential of a quadratic map over a finite field

I have some trouble with the definition of differential of a map over a finite field $\mathbb{F}_q$, where $q$ is a power of $2$. I have found that, given $f$ a quadratic map over $\mathbb{F}_q$, the ...
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1answer
15 views

Symmetric part of A contributes to quadratic form

In my statistics note, when it talks about quadratic forms, it goes on saying: "$x^tAx=\frac12x^t(A+A^t)x$ implies that only the symmetric part of A contributes to the quadratic form." I am having ...
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22 views

How to write the symmetric Hessian matrix for a log function?

Say f(x,y,z) = $y*ln(cos(z)+x^2)$ How would I write this as a Hessian matrix? Would this be the right step I need to take in order to calculate the second-order Taylor polynomial for the function?
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0answers
11 views

Representation of integers by sum of squares with linear constraints of a special form

I would like to know what integers $d$ can be written as a sum $d=\sum_{i=1}^N\sum_{j=1}^M a_{ij}^2 $ with $a_{ij} \in \mathbb{Z}$ and where the row and column sums of $a_{ij}$ are fixed $\sum_{i=1}^N ...
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0answers
69 views

Linear reformulation or approximation of a quadratic inequality set

Based on the useful comments I reformulated my problem - hopefully it's more clear now. Let $A,B \in \mathbb{R}^{d \times d}$ be symmetric positive-semidefinite matrices, $x \in \mathbb{R}^d$ and ...
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1answer
21 views

Lagrange Method of Quadratic Form the a Billinear Form

In the following question I have to present the bilinear form as sum of squares with Lagrange method. $$q(x_1,x_2,x_3,x_4)=2x_1x_4-6x_2x_3$$ However I don't know how I can do it here since none of ...
2
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2answers
71 views

Expectation of Univariate Quadratic Form under Multivariate Gaussian

Is there an obvious trick I am missing for solving the following integral: $$ \int_x P(y|x) W(x) (-x^TMx+2x^Tm -c)dx$$ Distributions are Gaussians and $M$ is symmetric. I know how to do the ...
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1answer
33 views

How many different values can $(x^2 + y^2, x^2 + 2y^2 )$ have mod 4?

It is known that if a prime number $p = x^2 + y^2 $ is equivalent to $p \equiv 1 \mod 4$. This is Fermat's theorem on the sum of two squares. My question is about the value of two simultaneous ...
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1answer
28 views

Are all “forms” linear maps from vector spaces to fields?

It seems that whenever we call something a "form": quadratic form, linear form, bilinear form, one-form, two-form, etc. it is always a linear (or perhaps not?) map from some vector space (or ...
2
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2answers
42 views

Why is the inner product of a quadratic form a quadratic form?

I was going through a derivation of the second derivative of the $\log \det X$ where $X$ is symmetric positive definite, I noticed that despite the second order approximation of log det is written as: ...
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2answers
47 views

Assume this equation has distinct roots. Prove $k = -1/2$ without using Vieta's formulas.

Given $(1-2k)x^2 - (3k+4)x + 2 = 0$ for some $k \in \mathbb{R}\setminus\{1/2\}$, suppose $x_1$ and $x_2$ are distinct roots of the equation such that $x_1 x_2 = 1$. Without using Vieta's formulas, ...