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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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 $x=Ty$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $L ... 1answer 182 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 ... 2answers 54 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 $I want to solve a linear quadratic programming problem and it requires that I put this expression in a quadratic form.$ ...
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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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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) ...
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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 ...
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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 ...
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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 ...
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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?
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Let A be symmetric matrix of order 3. Consider set $S=\{x\in\mathbb{R}^{3}:\; x^{T}Ax=a\}$. 1 (true). If S is unbounded for any $a\in\mathbb{R}$, then A is indefinite. 2 (false). If S is ...
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How to sketch $-3x^2 - 8xy + 3y^2 = 1$ [closed]

The equation is as follows: $$-3x^2 - 8xy + 3y^2 = 1$$ How to specify the axis of the given curve? How to as accurately as possible draw a curve defined by this equation?
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