Articles of quadratic forms

possible signatures of bilinear form on subspaces

Suppose to have been given the signature of a symmetric bilinear form on a finite dimensional vector space. Is there a general rule to get all the possible signatures of the restriction to subspaces of codimension 1? For instance, I know that if the signature is (-,+,+,+) all subspaces of codimension one have signatures (+,+,+), […]

Finding the parameters of an ellipsoid given its quadratic form

Suppose we have the quadratic form of an ellipsoid of the form $$ax^2 + by^2+cz^2+dxy+eyz+fxz+gx+hy+iz+j=0$$ I want to find centroid of the arbitrarily oriented ellipsoid, its semi-axes, and the angles of rotation. For the 2D case I found an answer here. I was wondering if someone can help me do the same for 3D.

Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

Sinha’s theorem can be stated as, excluding the trivial case $c = 0$, if, $$(a+3c)^k + (b+3c)^k + (a+b-2c)^k = (c+d)^k + (c+e)^k + (-2c+d+e)^k\tag{1} $$ for $\color{blue}{\text{both}}$ $k = 2,4$ then, $$a^k + b^k + (a+2c)^k + (b+2c)^k + (-c+d+e)^k = \\(a+b-c)^k + (a+b+c)^k + d^k + e^k + (3c)^k \tag{2}$$ for $k = […]

On the hessian matrix and relative minima

I’m asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. Suppose the hessian matrix of f is defined positive, prove that P is a strict relative minima of f. A similar proof has already been […]

An annoying Pell-like equation related to a binary quadratic form problem

Let $A,B,C,D$ be integers such that $AD-BC= 1 $ and $ A+D = -1 $. Show by elementary means that the Diophantine equation $$\bigl[2Bx + (D-A) y\bigr] ^ 2 + 3y^2 = 4|B|$$ has an integer solution (that is, a solution $(x,y)\in\mathbb Z^2$). If possible, find an explicit solution (involving $A,B,C,D$, of course). Motivation: I […]

Proving the multiplicativity of a binary quadratic form

Consider the set $S$ of all integers of the form $x^2+y^2+4xy$, where $x$ and $y$ are integers. How could one prove the set $S$ is closed under multiplication? I have tried the bashy brute force method, but to no avail. Perhaps someone could help?

Factorise $y^2 -3yz -10z^2$

How do I solve this question? I have looked at the problem several times. However, I cannot find a viable solution. I believe that it is a perfect square trinomial problem.

Ternary Quadratic Forms

Let $Q(x,y,z) = ax^2 + by^2 + cz^2$ where $a,b,c \in \mathbb{Z}_{\neq 0}$. Suppose that the Diophantine equation $Q(x,y.z) = 0$ has a non-trivial integral solution. Show that for any rational number $g$, there exist $x,y,z \in \mathbb{Q}$ such that $Q(x,y,z) = g$ I have trouble starting, any help will be appreciated! I know Legendre’s […]

Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can one prove that this vector is given by the eigenvector corresponding to the minimum eigenvalue of $\bf{A}$? I have a proof of my […]

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $$\text{minimize} \quad x’Ax \qquad \qquad \text{subject to} \quad x’Bx = x’Cx = 1$$ Where $A$ is symmetric and $B$ and $C$ are diagonal. Does anyone have a suggestion for an efficient way of solving this? Thank you.