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

learn more… | top users | synonyms

1
vote
1answer
26 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
33 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
7 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
121 views
+50

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} ...
0
votes
0answers
23 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
9 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
30 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
42 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
46 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 ...
2
votes
1answer
54 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
21 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
11 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
59 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
26 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 ...
1
vote
2answers
32 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
3answers
44 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
29 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
102 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 ...
2
votes
0answers
49 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 ...
0
votes
1answer
39 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 ...
6
votes
0answers
78 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
20 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
41 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
20 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 ...
0
votes
0answers
20 views

Eigenvectors of a quadratic form and iterative descent

I am interesting in using eigenvectors of a quadratic form to perform iterative steps to get the function value to a certain point. While other methods may be more common, my quadratic form is not ...
0
votes
1answer
24 views

How to calculate a vector of quadratic forms (matrix algebra)

Let's say I have a $N$ $d \times 1$ (column) vectors, $x_1, x_2,\dots,x_N$, and a $d \times d$ matrix, $A$ ($A$ can probably be symmetric and positive definite if it helps somehow). I want to end up ...
0
votes
0answers
12 views

Bicomplex quadratic forms

The quadratic form associated with a real matrix $Q$ and a real vector $\vec{x}$ is $\langle Q \vec{x}, \vec{x} \rangle$. The quadratic form associated with a real matrix $Q$ and a complex vector ...
1
vote
1answer
34 views

$LDL^T$ decompositon of a symmetric matrix and a matrix determinant expression for the lower triangular entries

Let $n$ be a positive integer, and let $M$ be an integral, symmetric, nonsingular matrix. As $M$ is nonsingular, there exists an $LDL^T$ decomposition such that $D = (d_j)$ is diagonal and ...
0
votes
0answers
44 views

Expected value of a bilinear form

I read many of the previous posts but I could not find my answer yet. Let $x \in \mathcal{C}(0,\sigma^2_x)$ and $y \in \mathcal{C}(\bar{y},\sigma^2_y)$ be two $N \times 1$ column vectors of i.i.d. ...
1
vote
0answers
49 views

Pell's equation and binary hyperbolic forms.

We define the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $0\neq c=a^2+b^2$. Is it true that $f$ is hyperbolic? In other word's is there any ...
3
votes
1answer
43 views

Pell,s equations and representation elements of $\mathbb Z_p$.

We defined the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $c\not\equiv 0\pmod{p}$ Is it true that $f$ is onto?
2
votes
0answers
15 views

Witt Groethendieck Ring splitting

I have a really basic question about the Witt Groethendieck ring of a field: In Lam's book, it says that $\hat{W}(F)/\hat{I}^2(F)$ depends only on the square classes of $F$, $\hat{W}/\hat{I}^2\cong ...
3
votes
1answer
66 views

Quadratic surfaces: Coordinates and radius( Non origin)

So I have a problem figuring out how to find the coordinates and radius to quadratic equations that are not in the form of $$(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 $$ Where the coordinates are going ...
0
votes
0answers
10 views

Most general conditions under which composition of binary quadratic forms is a group operation?

One direct way to compose two binary forms $f(x,y) = \langle a,b,c\rangle (=ax^2 + bxy + cy^2)$ and $f'(w,z) = \langle a',b',c'\rangle $ where $gcd(a,a',(b+b')/2) = 1$ and $b \equiv b' \pmod2$ is ...
2
votes
0answers
45 views

Alternative way to recognize that a real symmetric quadratic form is positive

A real symmetric quadratic form $g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j$ is positive (definite) if $g(x)>0$ for every $\mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0$. It is well known that a ...
3
votes
2answers
28 views

Does the set of possible values of a binary quadratic form determine the form

If two forms have the same range and discriminant, then due to reduction to a unique reduced form(that depends only on the 2 smallest numbers in the range of the form), we can conclude that the two ...
1
vote
2answers
40 views

Determine $A$ such that $Q=X'AX$ has chi-squared distribution.

Let $\boldsymbol X\sim N_n(\boldsymbol\mu,\boldsymbol\Sigma)$, where $\boldsymbol\Sigma$ positive-definite. I am trying to determine, in general, what form $\boldsymbol A$ (one example is ...
1
vote
1answer
26 views

Is a binary quadratic form (over any field) that represents both $\pm 1$ necessarily hyperbolic?

If a $2$-dimensional quadratic form over a field $\mathbb F$ that represents both $1$ and $-1$ necessarily hyperbolic? Edit: Assume that $\text{char } \mathbb{F} \neq 2$.
0
votes
1answer
28 views

Good book on random quadratic forms

I am studying some algorithms which are very much based on quadratic forms involving complex Gaussian Random vectors, something like this $ \vec{x}^* M \vec{x} $ where $x \in \mathbb{C}^{N \times 1}$ ...
0
votes
0answers
18 views

How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper. Equation: $min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$, where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...
1
vote
1answer
37 views

Is the minimiser of the quadratic form of a semi-bounded self-adjoint operator an eigenstate?

I am wondering whether the following fact, for which I know well the proof when $H$ is a Schroedinger operator (see Lieb-Loss, Analysis, Chapter 11), is also true in the general setting used below, ...
1
vote
3answers
74 views

Why is quadratic form defined via a symmetric bilinear form?

A typical definition of quadratic form goes like this: Let $B:V\times V \to F$ be a symmetric bilinear form. A function $Q : V → F$ defined by $Q(v) = B(v, v)$ is called a quadratic form. Why ...
2
votes
1answer
36 views

Spectral Theorem and Quadratic Forms

Let $A$ be a $3x3$ matrix which is not a diagonal matrix. Show that its eigenvalues are not all the same. Let $Q(x)$ be the corresponding quadratic form: show that $$\lim_{x\to 0} ...
0
votes
4answers
45 views

Complete the square in the form $(px+q)^2+r, p > 0$

I'm going over some completing the square questions and I need to express, in the form: $(px+q)^2+r, p > 0$ the quadratic equation is $16x^2-8x+11$ I know how to get it in the form $p(x+q)^2+r$ ...
1
vote
0answers
28 views

An inequality from $0-1$ matrices

Let $A\in\{0,1\}^{n\times n}$ of real rank $r$. Let $J$ be all one matrix. Denote $\underline{x}=(x_1,\dots,x_n)$, $\underline{y}=(y_1,\dots,y_n)$. It is clear we have ...
1
vote
2answers
34 views

Diagonalization of a symmetric matrix over algebraically closed field

Let $k$ be an algebraically closed field. Let $A$ be an $n \times n$ symmetric matrix with entries in $k$. Does it then follow that there exist eigenvectors of $A$ which form an orthonormal basis of ...
1
vote
1answer
25 views

linear algebra - Compute matrix associated to quadratic form.

We have a form: $Q: R^3\to R$, $Q(x) = 3x_1^2 + 3x_2^2 - 2x_1x_2 + 4x_1x_3 + 4 x_2x_3$, where $x = (x_1, x_2, x_3)$ is an arbitrary vector from $R^3$. The problem is to compute canonical form using ...