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

learn more… | top users | synonyms

1
vote
1answer
38 views

Tricky problem about quadratic forms

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 ...
0
votes
0answers
44 views

Hermitian form of a unitary space

If $V$ is a finite $n$-dimensional complex vector space with a hermitian form $h$ then $h$ is given by a hermitian matrix $A$ with the transformation law $P^{t}A\bar{P}$ where $P$ is an invertible ...
0
votes
1answer
28 views

Diagonalizing quadratic forms

I know how to diagonalize a given quadratic form using the Gaussian method. Though, I once read somewhere that there's another method which uses an augmented matrix, but I didn't go into details. I ...
2
votes
2answers
34 views

find real numbers so that a specific linear isometry exists

Consider the quadratic form $$f: \mathbb{R}^3 \to \mathbb{R}, (x_1, x_2, x_3) \mapsto 3x_1^2 - 3x_2^2 + x_3^2-2x_1x_3$$ I want to find $\lambda_1, \lambda_2, \lambda_3 \in \mathbb{R}$, so that there ...
8
votes
1answer
96 views

Why does the plot of the legendre symbol of $x^2 - y^2$ over a finite field look rectangular

The small top-left thing is a plot of the legendre symbol of $x^2 - y^2$ over $\Bbb F_{37}$. The thing in the middle is plot for $\Bbb F_{587}$. The thing on the right is a plot of the legendre ...
0
votes
1answer
17 views

Show that $|Q(a,b)| \leqslant M \cdot \max(|a|^2,|b|^2).$

Prove if $Q(x,y)=Ax^2+Bxy+Cy^2$ is a quadratic form, then the constant $M = |A|+|B|+|C|$ satisfies the property that $|Q(a,b)| \leqslant M \cdot \max(|a|^2,|b|^2)$ for every $a,b \in \mathbb{R}.$ ...
0
votes
0answers
23 views

Quadratic Form for Matrix with Sum

I just had a pretty basic question. Suppose you have the quadratic form: $$(u+x)^t V (u+x)$$ If you expand this out, you get something along the lines of $$u^t V u + x^t V x + u^t V x + x^t V u$$ ...
1
vote
0answers
38 views

Is the $\mathbb{Q}_2$- Space$ (\mathbb{Q}_2[\zeta], trace(cxy))$ hyperbolic?

I am working at a Problem for some time and it comes down to the question: Let $K:=\mathbb{Q}_2[\zeta]$ be a cyclotomic extention of the dyadic field $\mathbb{Q}_2$. For any $c \in ...
0
votes
0answers
25 views

Cycles of binary forms with coefficients in $F_q^*[T]$

I aim to study the binary forms $ax^2 + bxy + cy^2 = (a,b,c)$ where $a,b,c \in {F_q}[T]$, in particular those such that the discriminant $D = b^2 - 4ac \in F_q[T]$ has even degree and sign ${D} \in ...
0
votes
0answers
26 views

finding the axis of a hyperbolic cylinder

I have data (a lot of points x,y,t) which are modeled by a hyperbolic cylinder $t^2 = b_0+b_1x+b_2y+b_3x^2+b_4xy+b_5y^2$ I know that if i just make a set of 6 equations from it, and than randomly ...
0
votes
1answer
35 views

determining a linear isomorphism so that two quadratic forms become equivalent

Consider the matrix $$ G = \begin{pmatrix} 3 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & -3 \\ \end{pmatrix}$$ and the quadratic form $q: \mathbb{R}^3 \to \mathbb{R}$, given by $q(v) ...
0
votes
3answers
101 views

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?
1
vote
1answer
48 views

Products of quadratic forms

It is known that, if $x_1^2 + y_1^2 = c_1$ and $x_2^2 + y_2^2 = c_2$, then $(x_1x_2 + y_1y_2)^2 + (x_1y_2 - x_2y_1)^2 = c_1 c_2$ Is there a similar analogue for general quadratic forms $Q(x, y) = ...
1
vote
0answers
27 views

isometric quadratic spaces over a prime field

Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form. I want to show: either, $(V, \gamma)$ is ...
0
votes
1answer
305 views

Variance of a quadratic form

I am considering a variance of two forms: $ R(x) = (x-m)^\top A (x-m) + b^\top (x-m) + c $ $ R'(\Delta) = \Delta^\top A \Delta + b^\top \Delta + c $ where $x$ is a random variable of ...
3
votes
1answer
66 views

Second order derivation of Quadratic form

I would like to find the second order derivative of a Quadratic form. Assume we have a random complex column vector $x$ and a real constant value $C$. I am interested in computing the following: $$ ...
4
votes
0answers
46 views

maximal linear subspaces contained in the cone over the Clifford torus.

Forgot: this is about Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0 I was a little surprised to find that, in the cone $x^2 + y^2 = z^2 + w^2$ in $\mathbb R^4,$ there are infinitely many ...
5
votes
4answers
183 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
35 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
49 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\{ ...
0
votes
0answers
28 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
20 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
49 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
67 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 ...
1
vote
0answers
16 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
89 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 ...
2
votes
2answers
47 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
59 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
16 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
120 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) = ...
32
votes
2answers
1k 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
43 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 ...
2
votes
2answers
76 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 ...
2
votes
1answer
230 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: ...
0
votes
1answer
82 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 ...
1
vote
1answer
50 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
50 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 ...
0
votes
1answer
14 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 ...
3
votes
1answer
289 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
0answers
28 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 = ...
0
votes
0answers
18 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
74 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 ...
4
votes
3answers
47 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 $x^2 ≡ 4 \pmod p$? ...
1
vote
0answers
67 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 ...
1
vote
1answer
70 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
66 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
20 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: ...
2
votes
1answer
86 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
2answers
61 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
31 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 ...