# Tagged Questions

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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### $rad_{R}(V) = {0}$ if and only if $rad_{L}(V) = {0}$

How to prove that $rad_{R}(V) = {0}$ if and only if $rad_{L}(V) = {0}$? for V is a finite dimensional vector space? for V is infinite dimensional vector space?
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### Bounds on a quadratic form

I am currently in the middle of a proof where it would be nice to have some estimates on the size of a quadratic form. In particular, I am looking at $$x^TAx$$ where $A$ is "small" (in the analyst's ...
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### Integer solutions to $x^2-xy+y^2=1$

What are the integer solutions to $x^2-xy+y^2=1$? (I found the solution below while working on another problem, so I thought I'll add it to the knowledge base here.)
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### Quadratic forms with the same discriminant

Show that any quadratic form $ax^2+bxy+cy^2$ with discriminant $-4$ is equivalent to the form $x^2+y^2$, and any quadratic form of discriminant $-3$ is equivalent to $x^2+xy+y^2$. Here quadratic ...
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### Sesquilinear Forms

I was trying to solve some exercises related to sesquilinear forms: Let V be a C-vector space (C - complex numbers) Prove that the set $\mathcal{S}(V)$ of sesquilinear forms on V is a vector ...
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### Approximating a quadratic term in the constraint set as 2nd order Taylor expansion

I have an optimization problem in the following form: $$\min_{x,y} f(x)+g(y)$$ $$s.t.$$ $$Ax+h(y)=0$$ where $h(y)$ is a quadratic in $y$. Instead of solving this problem directly, $h(y)$ is ...
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### Maximization of quadratic form on a sphere [duplicate]

I have to following problem $$\max_{x}x^TAx+b^Tx\quad \mathrm{s.t.}\quad x^Tx\leq c,$$ where $A$ is real, symmetric and positive semi-definite. Firstly I tried to solve the problem with the KKT, but ...
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### Positiveness of a specified quadratic form

The condition of the positiveness of a ordinary quadratic form can be derived by getting the condition of positiveness of a square matrix, like ${v}^{T}{A}{v} \geq 0$ is equal to matrix $A \geq 0$ ...
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### Statement about composition of binary quadratic forms in “A Course in Computational Algebraic Number Theory”

On p.239 A Course in Computational Number Theory, Cohen writes "Although the group structure on ideal classes carries over only to classes of quadratic forms via the maps $\phi_{FI}$ and $\phi_{IF}$ ...
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### Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.

Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
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### Fermat's Challenge of composition of numbers

In his letter to Carcavi (August 1659), Fermat mentions the following challenge There is no number, one less than a multiple of $3$, composed of a square and the triple of another square. ...
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### Representations of some primes as $3x^2-4y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$p\equiv11\pmod{12}\longrightarrow p=3x^2-4y^2$$ Any help appreciated.
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### Hurwitz's matrix equations

I have a question about the proof of Hurwitz's 1-2-4-8 theorem about the sum of squares. I have consulted Chapter 1 of Rajwade's "Squares" book, notes by Keith Conrad, and notes by Daniel Shapiro. ...
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### How to solve similar with Transformation of a quadratic form into diagonal form?

Define $\color{red}{f=f(x),f'=f'(x)}$,where the derivative with respect to $x$ of a function $f(x)$ is denoted $f'(x)$. Now give six postive numbers $k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}$, and a ...
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### Representations of some primes as $x^2-2y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$p\equiv\pm1(\mod8)\longrightarrow p=x^2-2y^2$$ Any help appreciated.
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### Lagrangian of quadratic form with linear constraints

I am working through this paper, and I am confused as to how the author obtains the Lagrangian in equation (1.4) More specifically, in this paper, the authors present a problem: Minimize $x^T A x$ ...
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Consider the quadratic form $Q(v)=v^{t}Av,v=(x,y,z,w)$ where matrix $A$ is given by \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & \\ 0 & 0 & 0 &...
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### How to write a given element of the orthogonal group as a product of reflections

Let $V$ be a 3-dimensional vector space over a finite field $F$ of $q$ elements, where $q$ is an odd prime power. We know that the orthogonal group $O(V)$ is generated by reflections. How can a given ...
Assume $X$ to be a tall block-diagonal matrix where each block is a collumn vector. Assuming $X^+ = (X^H X)^{-1}X^H$ to be the pseudoinverse of the matrix $X$, find $X$ which maximizes $$y^H ( X X^+ ... 0answers 36 views ### Clifford Algebra Isomorphic to Exterior Algebra Let E be a vector space over a field k and Q be a quadratic form, that is,$$Q:E\to k$$such that$$Q(\lambda e)=\lambda^2Q(e)\forall\lambda\in k\,e\in E$$and such that P_Q:E^2\to k is ... 0answers 43 views ### Integral of a multivariate Gaussian distribution over quadratically separated partions Imagine in the space of \Re^n, the quadratic curve c: f(\mathbf{x}) = \mathbf{x}^TW\mathbf{x} + \mathbf{w}^T\mathbf{x} + w_0 (with W being a symmetric positive definite matrix, \mathbf{w} a ... 0answers 13 views ### Expectation of quadratic form of correlated variables Suppose U and V are two n\times 1 dependent vectors, in the sense that E\left( UV^{\prime}\right) \neq \mathbf{0}. For a given constant n\times n matrix A, is there is any simple way to ... 1answer 100 views ### What are numbers n such that a^2+nb^2 = c^2 and na^2+b^2 = d^2? Let n and a,b,c,d, be in the positive integers. I. For the system,$$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$then n is a congruent number. The sequence starts as n=5,6,7,13,14,15,20,21, and so ... 0answers 53 views ### Limit of ratio of quadratic forms Let Q_A,Q_B : \mathbb R^n \rightarrow \mathbb R be quadratic forms. Find a necessary and sufficient condition for \lim_{\vec x \rightarrow \vec 0} \frac{Q_A(\vec x)}{Q_B(\vec x)} to exist in the ... 1answer 22 views ### Quadratic form inequality implies matrix inequality? Suppose we have the following quadratic form:$$ x^T(t)(A^TP+PA)x(t)\le-x^T(t)Qx(t)\quad\forall t $$where P and Q are symmetric positive definite matrices and \dot x(t)=Ax(t). Why does the ... 1answer 13 views ### “Same range of values”, quadratic form transformation A quadratic form in the variables u_i is expressed as u'Du. Matrix T consists of the n characteristic vectors (of matrix D): T = [v_1\quad v_2\quad ...\quad v_n]. The following ... 2answers 30 views ### matrix trace bilinear form I'm wondering about this problem on bilinear forms : We have \phi : \mathbb{M_{n}(R)}*\mathbb{M_{n}(R)} \rightarrow \mathbb{R}$$(A,B) \rightarrow trace(AB)$$I've proved \phi is a bilinear form ... 0answers 19 views ### Reduction of Two Independent Random Variables in Quadratic Form Consider the n \times 1 random vector \mathbf{x} and the p \times 1 random vector \mathbf{y}. The vectors are independent of each other, and \mathbf{y} has an expected value of zero. I want ... 0answers 17 views ### Stabiliser of a non-isotropic 1-space in \Omega(n,q) Let n,q be odd, V be the n-dimensional vector space over \mathbb{F}_{q}, and consider the subgroup$$G=\Omega(n,q)=\{r_{v_{1}}r_{v_{2}}\dots r_{v_{k}} : k \textrm{ even }, \prod_{i=1}^{k}{(v_{...
I'm beginning with quadratics forms and I am wondering : Let $a$ be a real number and $q:\mathbb{R^4} \rightarrow \mathbb{R}$ given by $$(x,y,z,t) \rightarrow ax^2+2axy+y^2+4zt-at^2.$$ I would ...