Questions about quadratic forms in many variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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5
votes
4answers
131 views

How to solve an equation with $x^4$?

Today, I had this question on a Maths test about Algebra. This was the equation I had to solve: $$(1-x)(x-5)^3=x-1$$ I worked away the brackets and subtracted $x-1$ from both sides and was left with ...
3
votes
1answer
25 views

Why should the metrical groundform on a variety be a quadratic form?

I'm learning General Relativity and I can't understand why the distance function on space time is a quadratic form $$\textrm{d}s^2=g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}$$ I explain it through ...
2
votes
1answer
32 views

Mass of a wire: intersection of surfaces

So I got this mass problem to solve: Find the mass of the wire formed by the intersection of two surfaces whose density is $\phi=x²$ $\underset{C}\int \phi ds $ along the curve: $$ C:\left\{ ...
1
vote
2answers
37 views

Composition of binary quadratic forms as matrix operations

It is easy to see that any binary quadratic form $a^2 + 2bxy + cy^2$ is the same as $XAX^T$ where $X = [x, y]$ and $A = \begin{bmatrix}a & b\\b & c\end{bmatrix}.$ The composition of two ...
0
votes
0answers
25 views

Quadric form $ax^2-ay^2$ to $x^2-y^2$

Let $\mathbb{F}$ be field with characteristic $\ne 2$. And $q = a(x^2-y^2)$ - quadric form on $\mathbb{F}^2$. I want to prove that there is some basic such such $q = x'^2 - y'^2$. I have proved this ...
0
votes
1answer
19 views

Simplifying an unusual quadratic linear algebra expression

I came across the following expression when solving a maximisation problem. I have the following ingredients: Matrices $\Omega, P \in \mathbb{R}^{n \times n}$ Vector $t \in \mathbb{R}^n$ Also let ...
0
votes
0answers
11 views

question about isotropic subspaces

if $V$ is a complex vector space of dimension $2n$ and $Q$ a bilinear form over $V$, the definition of an isotropic subspace is the following: $$\Lambda:Q(\Lambda,\Lambda) \equiv 0$$. Suppose that ...
0
votes
1answer
28 views

Finding the diagonal representation of a quadratic form

Let $q:\mathbb{R}^n\to\mathbb{R}$ be a quadratic form: $$q(x_1,\dots,x_n)=\sum_{i=1}^{n} x_i^2+\sum_{1\leq i < j \leq n} x_i x_j$$ I must find the diagonal form of $q$. My attempt: I tried ...
2
votes
2answers
43 views

Quadratic matrix equation: ellipse of all solutions

Consider the following equation in $Z$: $$-2 (\pmb X^T Y)^T Z+Z^T(\pmb X^T \pmb X)Z = 0$$ where: $\pmb X\in\mathbb{R}^{n\times p}$ and $Y\in\mathbb{R}^n$ with $n>p$ are known and ...
1
vote
0answers
12 views

$\text{SO}(2,1)$ is isomorphic to the group of automorphism of $x^2+y^2-z^2$

I am not able to prove the following. Let $\mathbb{K}$ be a field with more than four elements and with characteristic different from $2$. Let $\mathcal{C} \subset \mathbb{P}_2(\mathbb{K})$ be the ...
2
votes
2answers
36 views

Are reducible Integral Binary Quadratic Forms equivalent?

By an integral binary quadratic form (IBQF for short) I mean an $$f(x,y) = ax^2 + bxy + cy^2$$ with $a,b,c \in \mathbb{Z}$. Note that I am not assuming that they are all coprime. Such an $f$ is said ...
0
votes
2answers
51 views

Compute $f_A(\lambda)$ without factoring cubic polynomial?

I'm given the following prompt: "Find the points closest to the origin on the surface defined by $x_1^2+2x_2^2+3x_3^2+x_1x_2+2x_1x_3+3x_2x_3=1$." What's the easiest way to compute the ...
1
vote
0answers
15 views

Linear Algebra quadratic forms (max and plot)

If I have $q(x)=x_1^2-x_1x_2-x_1x_3+x_2x_3$ How do I find the maximum value of $q(x)$ subject to the constraint $||x||=4$? I already know the max when $||x||=1$ since it is the eigenvalue, but I don't ...
0
votes
1answer
35 views

Reduction of quadratic forms

To reduce a quadratic form $q: \mathbb R^n \longrightarrow \mathbb R$, one can: $1)$ Use the method of Gauss. For instance, if we have: $q: \mathbb R^3 \longrightarrow \mathbb R$: $q(x_1,x_2,x_3) = ...
2
votes
1answer
30 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
2
votes
2answers
39 views

How can the level curves of a quadratic form be a pair of lines?

$x_1^2+4x_1x_2+4x_2^2=1\Rightarrow A\begin{pmatrix}1&2\\2&4\end{pmatrix}\Rightarrow ...
30
votes
2answers
784 views

Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = ...
1
vote
0answers
26 views

How to graph quadratic forms and label points closest to and furthest from the origin?

$x_1^2+4x_2^2+9x_3^2=1$ $x_1^2+4x_2^2-9x_3^2=1$ $-x_1^2-4x_2^2+9x_3^2=1$ I have to sketch these three surfaces and determine which are "bounded", which are "connected", and what the points ...
0
votes
1answer
75 views

How do i expand/simplify this quadratic (or quartic?) equation

I'm having trouble doing the following question, was wondering if anyone was able to lend a hand, would be greatly appreciated as i'm not too sure where to start or how to go about this. The ...
4
votes
2answers
167 views

Should isometries be linear?

Question Suppose $V$ is a (finite-dimensional) vector space over $F$ ($\operatorname{char }F\neq2$, due to user1551) equipped with a non-degenerate quadratic form $Q$, and $T$ is a ...
3
votes
1answer
144 views

Maximize the largest eigenvalue of a Hermitian matrix constrained by quadratic polynomials

I am looking for a method to maximize under $\mathbf{y}$ the largest eigenvalue of the following Hermitian matrix \begin{equation} S = \left [ \begin{array}{ccc} \mathbf{y}^{H}S_{11}\mathbf{y} ...
1
vote
1answer
31 views

If equation has integer solution it has solution for every prime p.

How to prove that if the equation in the form: $a_0 x_0^2 + a_1 x_1^2 + \dots + a_nx_n^2 = 0$ where $a_0, a_1, \dots , a_n \in \mathbb{Z}$, has an integer solution, then it has solution in ...
0
votes
1answer
34 views

Transforming quadratic forms, how is this theorem called?

In my textbook there is the following nameless theorem: Let $Q=\sum_{i,j=1}^n a_{ij}X_i X_j$ with $a_{ij}=a_{ji}\in K$ be a quadratic form in $n$ variables over a field $K$ not of characteristic ...
2
votes
1answer
64 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
0
votes
1answer
8 views

Matrix Quadratic Form

Say I have: $S Q S'$, where $Q$ is positive semi definite. Is there a quick way to see that this matrix is positive semi definite? I can see the resulting matrix being symmetric, but not immediately ...
0
votes
0answers
25 views

Squareclasses in transcendental extensions of the p-adics

Let $p$ be any prime and $k = \mathbb{Q}_p$. The structure of the square class group $k^*/k^{*2}$ is well known. It has four or eight elements depending on whether $p$ is odd or not. If we set $K = ...
1
vote
0answers
36 views

Albert- Algebras and Traceforms

Im new to the topic so this could be basic nonsense to you. Any Albert-Algebra $A$ has a trace map $T:A \rightarrow k$ and thus one can assign a quadratic form $q_A$ of rank $27$ by setting $q_A(x) = ...
0
votes
0answers
10 views

On quadratic forms representing integers

Proposition $4.1$ in http://www.dms.umontreal.ca/~andrew/Courses/Chapter4.pdf states quadratic form $F = ax^2 + bxy + cy^2\in\Bbb Z[x,y]$ represents $N\in\Bbb Z$ iff $d^2\equiv D\bmod 4N$ with some ...
2
votes
1answer
41 views

How to classify quadratic forms using their signature

I just did a question asking to classify the kind of curve of a given quadratic polynomial: $$0=3x^2+8xy+6y^2+12x+20y+17$$ I completed the square a few times and eventually (correctly) observed that ...
3
votes
3answers
43 views

If $p>3$ what are two solutions of $x^2 ≡ 4 \pmod p$?

Theorem used: "Suppose that $p$ is an odd prime. If $p \nmid a$, then $x^2 ≡ a \pmod p$ has exactly two solutions or no solutions." Question: If $p>3$ what are two solutions of $x^2 ≡ 4 \pmod p$? ...
3
votes
1answer
1k views

norm of a quadratic form

Suppose that $q$ is a quadratic form on $\mathbb{R}^n$, $q(x)=(x,Ax)$ say (or $q(x)=x^TAx$ if you prefer that notation). Then one could consider the quantity $$ \sup\{ \left|q(x)\right| : \left\| x ...
1
vote
0answers
47 views

Computing the matrix representation of the quadratic form $A \mapsto \text{tr}(A^2)$

Define the quadratic form $Q:\mathbb{R}^{2\times 2}\to\mathbb{R}$ by $$Q(A) = \text{tr}(A^2).$$ What is the matrix representation of this bilinear form with respect to the standard basis of ...
9
votes
1answer
3k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
2
votes
1answer
55 views

Small integral representation as $x^2-2y^2$ in Pell's equation

Let $k$ be a "representable" positive integer, in the sense that $k=|x^2-2y^2|$ for some integers $x,y$. Does it necessarily follow that $k$ can also be represented with small parameters, i.e. ...
0
votes
0answers
23 views

Inequalities of quadratic form

We know that the below inequality holds if A is positive definite ${\lambda _{\min }}\left( {{A}} \right){\left\| x \right\|^2} \le {x^T}{A}x$ or equivalently $\alpha{\left\| x \right\|^2} \le ...
0
votes
0answers
13 views

Quadratic form expressed with trace.

I am attempting to prove the following identity: $(x-a)A^T(x-a)=\text{tr}(Ax_cx_c^T)+n(a-\bar{x})^2 \text{tr}(A)$ where $x_c=(x-\bar{x})$ and the orders of the vectors are $n$. I got as far as: ...
1
vote
2answers
37 views

Quaternion order associated to a ternary quadratic form

I am a bit puzzled by the discriminant of a ternary quadratic form. According to Lehman 1992 and another related question, the discriminant of a ternary quadratic form is the half-determinant of its ...
0
votes
1answer
104 views

Can you explain this identity's secret with this Equation $n-th$ powers.

For $k = 0,1,2,3,4,5,6,7,8$, we have the equality, $$(-5)^k + (-119)^k + (-101)^k + (-215)^k + (-197)^k + 43^k + 157^k + 31^k + 217^k + 169^k\\ =\\ (-47)^k + (-161)^k + (-35)^k + (-221)^k ...
0
votes
1answer
28 views

Quadratic Forms and Associated Matrices

This might be a dumb question but when we write the matrix associated with a quadratic form, why does the matrix need to be symmetric in general? I'm asking because I'm thinking there isn't a unique ...
0
votes
0answers
27 views

Show that $f(\mathbf x)=e^{Q(\mathbf x)}$ is a convex function.

Here's the problem: Let $A$ be a positive definite symmetric matrix and let $Q(\mathbf x)$ denote the associated quadratic form on $\mathbb R^n$. Show that $f(\mathbf x)=e^{Q(\mathbf x)}$ is a convex ...
0
votes
3answers
47 views

Spectral Theorem / Quadratic Form Minimization Problem

Here is the problem: Let $A$ be an $n \times n$ symmetric matrix. Let $S = \{ \mathbf x \in \mathbb R^n : ||\mathbf x|| = 1 \} $ denote the unit sphere. Let $Q(\mathbf x) = \mathbf x ^TA\mathbf x $ ...
0
votes
1answer
30 views

Quadratic Functional Differentiability

I would like to solve the following: Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. Consider the quadratic functional $\Phi$ defined by: \begin{equation} ...
0
votes
1answer
18 views

Regarding the maximum eigen value

In a paper, the author removed the matrix $P$ and use the maximum eigenvalue multiplied by identity matrix , so is the following true? $$x^T P x \le x^T \bar\lambda(P) I x$$ where $x\in\mathbb ...
0
votes
1answer
42 views

zeros of $x^*Ax$, a quadratic form

The question hopefully says it all! We have a Hermitian matrix $A=A^* \in \mathbb{C}^n$ and a quadratic form: $f(x)=x^*Ax,~x\in \mathbb{C}^n$ We want to find the solution of $f(x) = x^*Ax = 0$ When ...
2
votes
0answers
53 views

Trying to prove a theorem on simultaneous diagonalisation of matrices

Let $B$ be a $n\times n$ real symmetric positive definite form, and $A$ be a $n\times n$ real symmetric form. There exists an orthogonal matrix O such that $O^TBO=I$ and ...
6
votes
0answers
83 views

Official name of Fermat's $x^2+3y^2$ theorem?

One of Fermat's more well-known claims is that for any prime number $p$, $p=x^2+3y^2\iff p\equiv 1\pmod 3$. Does this have an "official" name? (Another one which goes $p=x^2+y^2\iff p\equiv 1\pmod 4$ ...
0
votes
1answer
22 views

quadratic operator - image

i have a question regarding the image of a quadratic operator. Suppose I have $A\in\Re^{5\times 5}$ a symmetric matrix whose $a_{ij}$ entries are strictly positive and I am interested in the domain ...
2
votes
1answer
21 views

Partial derivative w.r.t. to the time of a time-dependend quadratic form

Suppose the quadratic form $$V(x(t), t) = \frac{1}{2} x^\mathsf{T}(t) P(t) x(t)$$ where $$x(t) \in \mathbb{R}^n,~P(t) \in \mathbb{R}^{n \times n},~\text{and}~P(t) = P^\mathsf{T}(t) > 0$$ ...
-1
votes
1answer
46 views

Two candidates attempt to solve a quadratic equation of the form x² +p x +q = 0 with wrong value. [closed]

Two candidates attempt to solve a quadratic equation of the form x² +p x +q = 0. One starts with a wrong value of p and finds the roots to be 2 and 6. The other starts with a wrong value of q and ...
1
vote
0answers
25 views

Maximizing ratios of quadratic forms with several norms

to maximize a ratio of quadratic forms, $(u^\top Mu) / (u^\top Ku)$, or a canonical correlation analysis ratio $(u^\top Rv) / [(u^\top Ku)^{1/2} \times (v^\top Lv)^{1/2}]$, is done straightforwardly ...