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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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 ...
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0answers
24 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 ...
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2answers
25 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 ...
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3answers
42 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 $ ...
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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} ...
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1answer
15 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 ...
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1answer
98 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 ...
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0answers
48 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 ...
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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 ...
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0answers
75 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 $p=x^2+y^2\iff p\equiv 1\pmod 4$ ...
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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 ...
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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 $$x(t) \in \mathbb{R}^n,~P(t) \in \mathbb{R}^{n \times n},~\text{and}~P(t) = P^\mathsf{T}(t) > 0$$ ...
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1answer
39 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 ...
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0answers
17 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 ...
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0answers
18 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 ...
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1answer
23 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 ...
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0answers
10 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 ...
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1answer
33 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 ...
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0answers
42 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. ...
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0answers
47 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 ...
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1answer
42 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?
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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
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1answer
65 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 ...
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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
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0answers
44 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
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2answers
25 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 ...
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2answers
39 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 ...
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1answer
24 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 $\text{char } \mathbb{F} \neq 2$.
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1answer
25 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 $x \in \mathbb{C}^{N \times 1}$ ...
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0answers
17 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(), ...
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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, ...
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3answers
68 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 ...
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1answer
35 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} ...
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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 $p(x+q)^2+r$ ...
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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 ...
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2answers
31 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 ...
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1answer
23 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 ...
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1answer
32 views

The value at the integral lattice of a positive definite quadratic form is discrete

A (real) quadratic form is a homogeneous plynomial of degree 2 (with real cofficients), in any number of variables. A quadratic form is positive definite if it is takes only nonnegative values. Now ...
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2answers
74 views

Cubes of the Form $3x^2\pm xy+5y^2$, with $x,y$ Coprime

Are there any cubes of the form $3x^2\pm xy+5y^2$, with x, y coprime ? Partly inspired by this question. I tried various computer searches of the form $|x|\le10^a$, $|y|\le10^b$ with $a+b=6$, all ...
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2answers
38 views

Definiteness of a Quadratic Form

The problem is as follows: For what values of c is the quadratic form $$Q(x,y) = 3x^2-(5+c)xy+2cy^2$$ positive definite, positive semidefinite, or indefinite? Ok. My approach was to find the ...
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0answers
25 views

A problem on unitary spaces

If $V$ is a unitary space with a hermitian form $\langle,\rangle$ and $v_1,...v_n$ are any $n$ vectors in $V$ then is it true that ${\rm det}(\langle v_i,v_j\rangle)\geq 0$? When does equality hold? ...
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0answers
43 views

Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 ...
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0answers
13 views

How to convert the following problem to the standard quadratic programming form

I have the following problem, which I guess it is QP, but I donot know how to convert it to the standard form. ${\rm minimize}\sum_{j=1}^{N}( y_j - (\sum_{i=1}^{N}p_iyi) )^2$ subject to $p_i\geq0$ ...
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0answers
12 views

Developping a long quadratic form like $(x-y-z-\mu)^t\Sigma^{-1}(x-y-z-\mu)$

Is there a way to some how develop a long quadratic form ? Maybe something like : $(x-y-z-\mu)^t\Sigma^{-1}(x-y-z-\mu)= (x-\mu)^t\Sigma^{-1}(x-\mu) - (y+z)^t\Sigma^{-1}(y+z)$ or is there another way ...
2
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2answers
123 views

What's so special about the form $ax^2+2bxy+cy^2$?

Binary quadratic forms are sometimes studied (e.g. by Gauss) in the form $$ax^2+2bx+cy^2$$ In other words, the second coefficient is assumed to be even, and the polynomial is assumed to be ...
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0answers
21 views

Reference about quadratic forms with discriminant 1

When I am reading Serre's $A$ $Course$ $In$ $Arithmetic$, Chapter 5, it deals with $quadratic$ $forms$ of some vector space $V$, which can be viewed as an extension of an $abelian$ $group$ $E$ of ...
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1answer
54 views

Equivalent quadratic forms

Two quadratic forms $$Q(x_1, x_2, \dots , x_n) \\ \text{ and } Q'(x_1, x_2, \dots , x_n)$$ are called equivalent $$\Leftrightarrow Q'(x)=Q(Tx), \text{ where } T \in M_n(K), \text{ invertible }$$ ...
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1answer
36 views

A quadratic form over $K-$vector space $V$

Let $K$ a field, $\operatorname{char} K \ne 2$. Definition: A quadratic form over $K$ is a homogeneous polynomial $Q(x_1, x_2, \dots , x_n) \in K[x_1, x_2, \dots , x_n]$ of degree $2$. If ...
2
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1answer
22 views

Clarification of some doubts: working with the restriction of a quadratic form

Let $q:\mathbb{R^3}\to\mathbb{R}$ such that $$q(x,y,z)=2x^2+3y^2+4xy-2xz.$$ I have to determine rank and signature of $q$, and so far it should be fine: I got $\operatorname{rk}(q)=3$ and ...
0
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2answers
32 views

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, how do I get the other root

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, then I know I can just take $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ and divide it by $x+1$ to get the other root. In a ...