Questions about quadratic forms in multiple variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.
0
votes
0answers
9 views
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 ...
0
votes
0answers
20 views
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 ...
2
votes
2answers
192 views
Calculus approach to solve this Quadratic equation problem
Both roots of the equation $$(x-b) (x-c) +(x-a) (x-c) +(x-a) (x-b) = 0$$ are always positive , negative or real. Prove your result.
By solving this equation I got $3x^2 - 2(a+b+c)x +ab + bc + ca ...
0
votes
1answer
36 views
Show that if a quadratic form is primitive then so are equivalent forms
A Quadratic form is primitive if the greatest common divisor of the coefficients of it's terms is 1.
I saw in number theory book that "it is easily seen that any form equivalent to a primitive form ...
0
votes
0answers
25 views
what is a ordinally quadratic function？
A function is ordinal equivalent to another means there exist a (unique) monotonic transformation between wiki definition of ordinal utility. I am a little confused, a function is ordinally quadratic ...
0
votes
0answers
10 views
Maximize function symbolically
I have the following expression:
$$
\sum_{i,j=1}^n\rho_{ij}^2-\frac{2}{n}\sum_{i=1}^n\left(\sum_{j=1}^n\rho_{ij}\right)^2 +\frac{1}{n^2}\left(\sum_{i,j=1}^n\rho_{ij}\right)^2
$$
My goal is to ...
7
votes
2answers
112 views
Solving a quadratic 9-equation system
I need to solve the following system:
$$\begin{cases}
A^TA=B &(1)\\
A\vec{x}=\vec{y} &(2)\\
\end{cases}
$$
I need $A$, given $B$, $\vec{x}$ and $\vec{y}$.
$A$ and $B$ are both 3-by-3 ...
0
votes
0answers
18 views
Completing a multivariate square
A well-known trick when analyzing quadratic polynomials $P=ax^2+bx+c$ is to complete the square: P can be written as
$$P=\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a^2}.$$
I have been trying to ...
1
vote
1answer
60 views
Transforming Diophantine quadratic equation to Pell's equation
I have been discussing the fastest and most efficient ways of solving QDEs in a separate question record (Alternative method to solve quadratic Diophantine equations). However, as suggested by ...
0
votes
1answer
13 views
Factoring binary quadratic form in two second order polynomials
I have a binary quadratic form in $N$ and $D$, $AD^2 + BND + CN^2$, where $A$, $B$, and $C$ are real coefficients and $N$ is a second order polynomial of $x$ with real roots $\lvert r \rvert <1$ ...
-1
votes
1answer
20 views
Quadratic forms matrices
Let $$Q(x,y,z) = – 2x^2 + 6xy + 8y^2 + z^2.$$ Find the symmetric matrix associated with this
quadratic form. Use the determinant method to determine whether the quadratic form is
positive definite, ...
2
votes
0answers
50 views
On the integer solutions to $u^2+163v^2=w^3$ and others
It seems the solution of,
$$u^2+dv^2 = w^3\tag1$$
involves the class number $h(d)$. Assume $\gcd(u,v)=1$.
Q: For which $\color{red}{prime}\; d$ is the complete solution of $(1)$ in the integers ...
6
votes
0answers
37 views
Why do isotropic spaces deserve their name?
Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions?
From my experience so far, ...
1
vote
1answer
60 views
Maximization of vector norm under a quadratic convex inequality constraint
I need help for the following problem:
$$
\max_x x^Tx\quad \mathrm{s.t.}\quad x^TAx+b^Tx\leq c,
$$
where A is symmetric, square and positive semidefinite, c is a real scalar and b is a real vector.
...
4
votes
2answers
97 views
Unable to find solution for $a^2+b^2-ab$, given $a^2+b^2-ab$ is a prime number of form $3x+1$
I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$.
Now I am having list of prime numbers of form $3x+1$ ...
0
votes
1answer
57 views
Regarding the factorization $a^2+3b^2 = cd$.
Let $a,b,c,d$ be positive integers, with $\gcd(c,d)=1$, such that
$$a^2+3b^2=cd.$$
By well-known classical results, we have that $c$ and $d$ are both of the form $u^2+3v^2$.
QUESTION: Is it valid to ...
0
votes
1answer
26 views
Lower boundary of quadratic form
I have a quadratic form $x^TAx$ where $x$ is an $n \times 1$ vector and $A$ is a positive definite matrix in the sense that it has only positive eigenvalues. Am I right to say that $||x^TAx|| \ge ...
0
votes
1answer
55 views
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 $A \geq 0$ ...
3
votes
0answers
34 views
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 $\phi_{IF}$ ...
2
votes
1answer
30 views
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 ...
7
votes
3answers
62 views
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.
...
0
votes
2answers
43 views
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.
0
votes
1answer
17 views
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. ...
3
votes
1answer
72 views
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 ...
0
votes
1answer
31 views
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.
0
votes
0answers
10 views
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 $x^T A x$ ...
1
vote
1answer
41 views
Quadratic form.
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 ...
6
votes
1answer
101 views
On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$
The problem is to find four integers $a,b,c,d$ such that,
...
2
votes
1answer
29 views
Relation between Pfister forms in W(Q)
For $n$ a nonzero integer, is it possible to have a relation
$$ n\langle\langle-1,-1\rangle\rangle=\sum_i n_i\langle\langle a_i,b_i\rangle\rangle$$ in $I^2(\mathbb Q)\subset W(\mathbb Q)$ (or maybe ...
2
votes
1answer
63 views
How canonical is Gauss's law of composition of forms
Gauss defined the composition of binary quadratic forms $f$ and $g$ to be another binary quadratic form $F$ such that there exist integral quadratic forms
$$ \begin{align} r(x_0,x_1,y_0,y_1) &= ...
1
vote
1answer
40 views
How to write a given element of the orthogonal group as a product of reflections
Let $V$ be a 3-dimensional vector space over a finite field $F$ of $q$ elements, where $q$ is an odd prime power. We know that the orthogonal group $O(V)$ is generated by reflections. How can a given ...
0
votes
0answers
20 views
Polarization of a quadratic Hermitian form
Suppose $f$ is a conjugate-symmetric sesquilinear form over a vector space $V$ defined over $F=\mathbb{F}_{q^2}$, i.e. $f:V\times V \rightarrow F$ such that
$$f(\lambda u + v, w) = \lambda f(u, w) + ...
0
votes
0answers
9 views
Maximizing $y^H ( I - X pinv(X) ) y $ with respect to matrix $X$. How hard can it get?
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
$$y^H ( X X^+ ...
1
vote
0answers
30 views
Clifford Algebra Isomorphic to Exterior Algebra
Let $E$ be a vector space over a field $k$ and $Q$ be a quadratic form, that is, $$Q:E\to k$$ such that $$Q(\lambda e)=\lambda^2Q(e)\forall\lambda\in k\,e\in E$$
and such that $P_Q:E^2\to k$ is ...
3
votes
0answers
42 views
Integral of a multivariate Gaussian distribution over quadratically separated partions
Imagine in the space of $\Re^n$, the quadratic curve $c: f(\mathbf{x}) = \mathbf{x}^TW\mathbf{x} + \mathbf{w}^T\mathbf{x} + w_0$ (with $W$ being a symmetric positive definite matrix, $\mathbf{w}$ a ...
0
votes
0answers
11 views
Expectation of quadratic form of correlated variables
Suppose $U$ and $V$ are two $n\times 1$ dependent vectors, in the sense that
$E\left( UV^{\prime}\right) \neq \mathbf{0}.$ For a given constant $n\times n$ matrix
$A,$ is there is any simple way to ...
4
votes
1answer
86 views
What are numbers $n$ such that $a^2+nb^2 = c^2$ and $na^2+b^2 = d^2$?
Let $n$ and $a,b,c,d,$ be in the positive integers.
I. For the system,
$$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$
then $n$ is a congruent number. The sequence starts as $n=5,6,7,13,14,15,20,21,$ and so ...
1
vote
0answers
51 views
Limit of ratio of quadratic forms
Let $Q_A,Q_B : \mathbb R^n \rightarrow \mathbb R$ be quadratic forms. Find a necessary and sufficient condition for $\lim_{\vec x
\rightarrow \vec 0} \frac{Q_A(\vec x)}{Q_B(\vec x)}$ to exist in ...
0
votes
1answer
18 views
Quadratic form inequality implies matrix inequality?
Suppose we have the following quadratic form:
$$
x^T(t)(A^TP+PA)x(t)\le-x^T(t)Qx(t)\quad\forall t
$$
where $P$ and $Q$ are symmetric positive definite matrices and $\dot x(t)=Ax(t)$. Why does the ...
0
votes
1answer
12 views
“Same range of values”, quadratic form transformation
A quadratic form in the variables $u_i$ is expressed as $u'Du$.
Matrix $T$ consists of the n characteristic vectors (of matrix $D$): $T = [v_1\quad v_2\quad ...\quad v_n]$. The following ...
0
votes
2answers
21 views
matrix trace bilinear form
I'm wondering about this problem on bilinear forms :
We have $\phi : \mathbb{M_{n}(R)}*\mathbb{M_{n}(R)} \rightarrow \mathbb{R}$ $$(A,B) \rightarrow trace(AB)$$
I've proved $\phi$ is a bilinear form ...
0
votes
0answers
16 views
Reduction of Two Independent Random Variables in Quadratic Form
Consider the $n \times 1$ random vector $\mathbf{x}$ and the $p \times 1$ random vector $\mathbf{y}$. The vectors are independent of each other, and $\mathbf{y}$ has an expected value of zero. I want ...
0
votes
0answers
15 views
Stabiliser of a non-isotropic 1-space in $\Omega(n,q)$
Let $n,q$ be odd, $V$ be the $n$-dimensional vector space over $\mathbb{F}_{q}$, and consider the subgroup
$$G=\Omega(n,q)=\{r_{v_{1}}r_{v_{2}}\dots r_{v_{k}} : k \textrm{ even }, ...
1
vote
1answer
37 views
Degenerate quadratic form
I'm beggining with quadratics forms and I ma wondering :
Let $a$ be a real number and $q:\mathbb{R^4} \rightarrow \mathbb{R}$
$(x,y,z,t) \rightarrow ax^2+2axy+y^2+4zt-at^2$
I would like to know for ...
0
votes
2answers
34 views
Maximums on Quadratic Functions [closed]
How do you find the maximum of a quadratic function? Specifically,
$R(x) = -4x^2 + 4000x$
0
votes
1answer
30 views
A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots
I'm looking for a quadratic form of the form $q(x,y)=ax^2 + bxy + cy^2 \in F[x,y]$, where $F$ has characteristic 2, and $q(x,y)$ has no roots besides the obvious one, $x=y=0$. I've proved the case of ...
1
vote
3answers
32 views
A nontrivial solution to the quadratic form $x^2 - xy + y^2$ over the finite field $𝔽_p$ with $p ≡ 1 \pmod3$ a prime
I'm trying to prove that when $p ≡ 1 \pmod3$ is a prime, $p$ is reducible over the Eisenstein integers, and I've gotten to the point where, provided $p\,|\,u^2 - u + 1$ for some integer $u$, then $p$ ...
1
vote
1answer
44 views
Is the squared euclidean norm a measure for the distance of two points?
I like to prove that a measure for the distance $d$ of two points $\vec a$ and $\vec b$ in $R^N$ is given by the squared euclidean norm
$$d^2= \sum^N_j (a_j - b_j)^2 $$
So far I was able to show ...
1
vote
1answer
33 views
Vertex Form of Parabola - Why does it work?
Recently, I have been trying to plot parabolas of quadratic equations.
First, I have to convert them to vertex form and then we can easily plot them.
This makes me wonder why the vertex form of a ...
2
votes
0answers
51 views
Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$
Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$.
Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
