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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1answer
65 views

When are we allowed to match coefficients?

Related to this answer: Find k in $(1−2k)x^2−(3k+4)x+2=0$ given facts about the roots. In the partial fraction decomposition of $\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$, we have: $0x + ...
1
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2answers
78 views

Find k in $(1-2k)x^2 - (3k+4)x + 2 = 0$ given facts about the roots.

The exact instruction in my book is: A quadratic equation $(1-2k)x^2 - (3k+4)x + 2 = 0$ is given. Find the value of k for each of the following conditions. (I got 46a and 46b) (c) A ...
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0answers
23 views

Normal forms of a quadratic form of two variables.

If we are given a form $Q(x,y) = ax^2 + 2bxy +cy^2$, then, using a rotation (or a linear change in coordinates) we may eliminate the $xy$ cross term to obtain the following form: $Q(X,Y) = AX^2 + ...
0
votes
1answer
47 views

Primes of special form

Are there infinitely many primes $p$ of form $$2^k+a^2=p^2<2^{k+2}$$ where $a\in\Bbb N$? Which primes are known to be of such form? An example is $16+3^2=5^2$. This is the only one I could find.
1
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1answer
56 views

Sum of squares as Primes Class Field Theorem statements

We know that every prime $1\bmod 4$ can be written in an unique way as $a^2+b^2$ form where $a,b\in\Bbb N$. Is there a comprehensive list of other statements of form "every prime $d\mod r$ can be ...
2
votes
2answers
54 views

Quadratic form has non-trivial zero?

For each of the following quadratic forms, determine whether the form has a non-trivial zero (we do not need to exhibit it): $f(x, y, z) = 2x^2 + 3y^2 - 6z^2$; $g(x, y, z) = 2x^2 + 3y^2 - ...
2
votes
4answers
108 views

Is the matrix $A$ positive (negative) (semi-) definite?

Given, $$A = \begin{bmatrix} 2 &-1 & -1\\ -1&2 & -1\\ -1& -1& 2 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Define the ...
0
votes
1answer
57 views

How many ways can a quadratic form represent a prime?

Given $a,b,c,p\in\Bbb N$ with $b^2-4ac<0$ and $p$ is a prime with $\bigg(\frac{b^2-4ac}p\bigg)=1$, how many solutions $(x,y)\in\Bbb Z^2$ are there to $$ax^2+bxy+cy^2=p?$$
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1answer
50 views

How to transform the quadratic form of an ellipse to a circle

Consider the ellipse $x^TPx\le a$. I would like to transform (the quadratic form of) this ellipse into a circle $y^T\begin{pmatrix}1&0\\0&1\end{pmatrix}y\le b$ via a coordinate transform ...
1
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1answer
34 views

quadratic form in hilbert space and Gram matrix

We are in Hilbert space $L^2$ we are given a subspace of dimension K as $$ V=\{ g_k,1 \le k \le K \} $$ everything that folows is defined on $V$ we define map $$ x \mapsto Q(x):= \sum_{k=1}^{K} ...
0
votes
1answer
53 views

Quadratic form in Hilbert space associated with orthogonal projection operator

we are in Hilbert space $L^2 $ and we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
0
votes
0answers
23 views

Moments of quadratic forms

$x=(x_1,...,x_T)'$ is a $T\times1$ random vector, where $x_t, t=1,..., T$, is a stationary process with mean zero and finite fourth moments. $A$ is a $T\times T$ symmetric constant matrix. How to find ...
0
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4answers
74 views

Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.

I try to transform Transform $$f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$$ to a diagonal form. I can do it using eigenvalue, but when I directly complete the square to find its ...
3
votes
2answers
94 views

Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
0
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0answers
59 views

Simultaneously diagonalise two real quadratic forms

I would like to simultaneously diagonalise the quadratic forms $A=2x^2+3y^2+3z^2-2yz$, and $B=x^2+3y^2+3z^2+6xy+2yz-6zx$. Of course there's a theorem saying this is possible and I followed the ...
2
votes
1answer
64 views

How to compute the matrix $S$ in Sylvester's law of inertia

Sylvester's law of inertia states that for any symmetric matrix $A$ there exist an invertible matrix S such that, $S^T A S = D$, where $D$ is a diagonal matrix which has only entries 0, +1 and −1 ...
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2answers
22 views

Can I find a vector with entirely nonzero entries for the following quadratic form to evaluate to zero?

I have a square, symmetric matrix $M$, of size at least $2\times 2$, with diagonal entries equal to $1$ and off-diagonal entries equal to $\pm 1$. Let the entries of $M$ be such that it is indefinite. ...
1
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1answer
48 views

Omission in Jacobson's BAI regarding extension of isometries.

Suppose $V$ is a finite dimensional vector space over a field of characteristic $\neq 2$ equipped with a nondegenerate quadratic form $Q$. Witt's cancellation theorem says that if $U_1,U_2$ are ...
3
votes
2answers
108 views

Multivariate Gaussian integral of ratio of quadratic forms

Given two real symmetric matrices $M,S$ is there a known answer for the Gaussian integral $\int d^Nz\frac{z^TMz}{z^TSz}$ where the integration is over N-dimensional Gaussian variable $z\sim ...
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3answers
172 views

reference for linear algebra books that teach reverse Hermite method for symmetric matrices

January 13, 2016: book that does this mentioned in a question today, Linear Algebra Done Wrong by Sergei Treil. He calls it non-orthogonal diagonalization of a quadratic form, calls his first method ...
2
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2answers
117 views

Find the transitional matrix that would transform this form to a diagonal form.

Let the quadratic form $F(x,y,z)$ be given as below $F(x,y,z)=2x^2+3y^2+5z^2-xy-xz-yz$ Find the transitional matrix that would transform this form to a diagonal form. I got the symmetric ...
0
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0answers
41 views

Minimizing a quadratic form with orthogonality constraints

Suppose $A$ is an $n$-by-$n$ symmetric matrix, and I want to find $x_{1}$ and $x_{2}$ that maximize $x_{1}^{T} A x_{1} + x_{2}^{T} A x_{2}$ subject to the constraint that $x_{i}^{T} x_{j} = ...
0
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0answers
39 views

Find Isotropic vectors that form a basis

I have this question: let $(E,\langle,\rangle)$ an inner product space with dimension $n$ and $u$ a symmetric linear transformation and we define a quadratic form $q$ by $$\forall x\in E,\quad ...
0
votes
2answers
30 views

Solve the algebraic expression for a, b, and c of the function x

I am trying to solve for $a$, $b$, and $c$ in the expression below, but I have found that the way I tried to solve it is convoluted and did not work out. I believed that by solving for x, I would be ...
4
votes
1answer
133 views

Reducing a pair of indefinite quadratic forms to the canonical form

Assume $A, B$ being a pair of symmetric matrices over reals. Let $$ \varphi_1(x) = (x, Ax)\\ \varphi_2(x) = (x, Bx). $$ There's a well-known result that if $A > 0$ then the pair of forms can be ...
6
votes
4answers
162 views

Parametrization of $x^2+ay^2=z^k$, where $\gcd(x,y,z)=1$

$x,y,z$ be three coprime integers, $a \in \mathbb{Z}>0$ and $k$ an odd integer. How do I find all the non-trivial solutions of the diophantine equation? $$x^2+ay^2=z^k$$ Does the method which ...
8
votes
0answers
72 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
5
votes
2answers
145 views

How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...
1
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1answer
48 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
2
votes
1answer
85 views

Quadratic form as generalized distance?

In the book A Linear Systems Primer (by Antsaklis and others), they first mention squared distance of a point x from the origin: $$x^{T}x = ||{x}||^2$$ which represents the square of the ...
0
votes
2answers
50 views

Can this expression be made into a quadratic form?

Can this expression be made into a quadratic form: $ a x_t -\gamma {x_t}^2 $ I want to solve a linear quadratic programming problem and it requires that I put this expression in a quadratic form. $ ...
0
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0answers
33 views

How to do linear quadratic dynamic programming with non homogeneous quadratic equation

I am not well versed on matrix algebra and linear quadratic programming. I am wondering if it is possible to make a non-homogeneous equation into a homogeneous one. I need to make the following ...
0
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1answer
18 views

Almost universal integer quadratic forms

This question is inspired by the 15-theorem. For any nonnegative integer k, define a k-universal integer quadratic form to be a form that represents all but k positive integers. So, universal forms ...
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0answers
42 views

Question about Mumford's article

I'm reading the following article by Mumford speaking about theta characteristic. Mumford's article I'm trying to understand the definition af the quadric form $q$ on page 184. Here my questions: 1) ...
0
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0answers
38 views

how to determine a biquadratic form is positive-definite

A biquadratic form $\sum_{i,j,k,l}b_{i,j,k,l}x_{i}x_{j}y_{k}y_{l}$, how to determine whether it is positive-definite? A necessary and sufficient condition? In fact, I have a matrix $B=\sum_{1\leq ...
3
votes
0answers
37 views

Quadratic form and matrix

We know quadratic form $f(x_1,x_2)= a_{11} x_1^2 + 2 a_{12} x_1 x_2 + a_{22} x_2^2$ is non-negative for all $x_1,x_2 \in \mathbb{R}$ iff matrix $(a_{ij})_{2 \times 2}$ is semi-positive defined. My ...
0
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0answers
39 views

Complexity of computing a posiform of a quadratic pseudo-boolean function

I am reading the chapter 13, Pseudo-Boolean functions, of Boolean Functions: Theory, Algorithms, and Applications by Crama et. al. In section 13.2, the authors introduce the idea of Posiform. The ...
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2answers
26 views

graph quadratic form and find the equation of asymptotes

So I had this quadratic form that need to be graphed showing both original and new axes. And I also need to find out the equation of asymptotes. $$ \left\{ \begin{aligned} ...
0
votes
1answer
36 views

Maximization of quadratic form over complex unit cube

I am trying to find the maximum of a hermitian positive definite quadratic form $xQx^H$ (where $Q=Q^H$ and all eigenvalues of $Q$ are non-negative) over the complex unit cube $|x_i|\leq 1$, ...
1
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0answers
42 views

Differentiating quadratic form containing vector raised to powers elementwise, can I avoid Hadamard notation?

Say $\mathbf{M}$ is a symmetric, p.d. 2x2 matrix, and $\mathbf{x}$ is a 2x1 vector. The familiar quadratic form is of course given by: $A=\mathbf{x'}\mathbf{M}\mathbf{x}$ (where $A$ is a scalar), and ...
0
votes
1answer
41 views

existance of a solution to quadratic form equation

Let $\lambda$ is an unknown scalar and; $Q=Q_1 - \lambda*Q_2$ where $Q_1, Q_2$ are $NxN$ positive symetric matrices, $B=B_1 - \lambda*B_2$, where $B_1, B_2$ are $Nx1$ vectors, $m=m_1 - ...
0
votes
2answers
46 views

Minimum quadratic form value within a line?

If I have $x\in R^n , C\in R^{m\times n}, d\in R^m$, $m<n$, then $Cx=d$ is a linear manifold. And $P\in R^{n\times n}$, $P>0$, the quadratic form is $y=x^TPx$ Is there an analytical expression ...
3
votes
1answer
93 views

Combining sums and/or differences of squares

I'd like to combine a sum of as many squares as possible into a sum of as few squares as possible. The signs of the squares doesn't matter. For example, the Brahmagupta-Fibonacci Identity combines a ...
1
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0answers
34 views

Semidefinite relaxation of Quadratic equation

I have read in various papers that we can write a Quadratic equation with symmetric matrix as a linear programming problem. For example $$f(x)= x^T*Q*x + c$$ where Q=[2 0;0 3]; Now we can write ...
1
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0answers
25 views

Reduction of positive definite binary quadratic forms over congruence subgroups

Let $\Gamma_0(N)$ be a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and $Q(x,y)$ be a positive definite binary quadratic form with leading coefficient $a$ divisible by $N$. Can someone give me a ...
2
votes
2answers
92 views

Regular Quadratic Space - isotrope vector

I am currently trying to solve the following exercise: Show that every regular quadratic space of finite dimension $E$ that contains at least one isotrope vector, has a basis consisting only of ...
0
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0answers
11 views

non-uniform arithmetic lattice in a semisimple algebraic group over a local field of positive characteristic

Say I'm considering the group of rational points $G(k)$ where $G$ is the special orthogonal group for the quadratic form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}-x_{5}^{2}$ and $k$ is a ...
0
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0answers
21 views

example of a quadratic form

Would someone be able to tell me an example of a quadratic form defined on a five-dimensional vector space $V$ over a non-archimedean local field $k$ of positive characteristic (not equal to two, say) ...
1
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0answers
26 views

What is binary norm of quadratic fields of sum of two squares such that one of them is necessarily even like $a^2 +4b^2?$

I am trying to simplify an expression which I have reached, suppose a number can be represented in the form of $D=a^2 + 4b^2$. What is binary norm of $D$, or how else can it be represented?
0
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0answers
27 views

meaning of index of a quadratic form for a field of positive characteristic

In "Classification of Algebraic Semisimple Groups" in "Algebraic Groups and Discontinous Subgroups: Procedings of Symposia in Pure Mathematics, Volume IX", Jacques Tits speaks of the index of a ...