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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13 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 ...
2
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2answers
13 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
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1answer
19 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
18 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
16 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(), ...
2
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2answers
139 views

Quadratic Forms and Congruences

How does one prove (the non-trivial direction) that, for $n \in \mathbb{N}$, $x^2 + y^2 + z^2 = n$ solvable $\iff$ $x^2 + y^2 + z^2 \equiv n\ (m)$ solvable for all $m$? In particular, is there a ...
1
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1answer
34 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
57 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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2answers
17 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 ...
2
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1answer
33 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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1answer
31 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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1answer
2k 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. ...
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4answers
43 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
26 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
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1answer
20 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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2answers
65 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
37 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
24 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
19 views

How I can find index of inertia?

How I can find index of intertia of matrix: $ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $ and matrix: $ \begin{bmatrix} 0 & 1\\ 1 & 0 & 1\\ ...
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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
10 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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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
9 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 ...
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0answers
28 views

Quadratic form - non-degenerate

(The order of a quadratic form is defined to be the order of the matrix $A$) Definition: $Q(x_1, x_2, \dots , x_n)$ is called non-degenerate $\Leftrightarrow (a) $A=$invertible (b) At each $v \in ...
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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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0answers
10 views

Two modules with the same index have the same discriminant

In Serre's book $A$ $Course$ $in$ $Arithmetic$, it writes the following: Let $n=4k$, $k$ be a positive integer, $V=\mathbb Q^n$ be a $\mathbb Q$-vector space, endowed with the standard bilinear form ...
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2answers
27 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
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1answer
52 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
31 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
21 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 ...
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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 ...
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0answers
28 views

What is a “supplementary subspace”?

Let $Q$ be a quadratic form of vector space $V$ over a field $k$ with characteristic $\neq 2 $, $V^{0}$ be its orthogonal complement. If $U$ is a supplementary subspace of $V^0$ in $V$, then $V = ...
3
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1answer
152 views

Any integer can be written as $x^2+4y^2$

If $n$ is a positive integer with $(n,8)=1$ and $-4$ is square $mod$ $n$ then $n$ can be written in this form: $n=x^2+4y^2$. I was using that there are x, y integers satisfying $x^2+4y^2=kn$ where ...
5
votes
2answers
107 views

How to reduce a quartic form to a quadratic form with equal roots

Given a polynomial in $n$ variables of the form $$P(x_1,x_2,\dots,x_n)=\left(\sum_{i,j}a_{ij}x_ix_j+\sum_{i}b_{i}x_i+c\right)^2$$ is there a way to find a polynomial also in $n$ variables of degree ...
4
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0answers
29 views

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $\min_{x} (x'Ax)$ subject to $x'Bx = x'Cx = 1$. Where A is symmetric and B and C are diagonal. Does anyone have a suggestion for an ...
3
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0answers
32 views

(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal ...
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0answers
43 views

Show Equivalence of Binary Quadratic Forms

I've been stuck on these two problems from my problem set for quite a while. Any help would be appreciated! 2)Suppose that $ax^2 + bxy + xy^2$ is equivalent to $Ax^2 + Bxy + Cy^2$. Show that $gcd ...
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1answer
20 views

The set of positive definite forms in the space of quadratic forms

Let $u_1,...,u_k\in\mathbb{R}^n$ such that there is a non-zero quadratic form $Q$ satisfying $Q(u_i)=1$ for all $i=1,...,k$. Is there a positive definite quadratic form satisfying the same equations? ...
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1answer
39 views

A question about minors of matrices

Let $B_{\bar{i}\bar{i}}$ denote the remnant of a square matrix $B$ after its $i^{th}$ row and $i^{th}$ column have been removed. Now given any vector $v$ is there some natural relation between the ...
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0answers
43 views

A question on constructing subharmonic rational functions

Suppose three functions $x\leq y\leq z$ satisfy $x+y+z=0$ and \begin{equation*} \begin{split} \nabla x=&2x\vec a-y\vec b-z\vec c,\\ \nabla y=&-x\vec a+2y\vec b-z\vec c,\\ \nabla z=&-x\vec ...
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2answers
57 views

Solve the simultaneous equations $(x+y)^2+3y^{2}=7$ and $x+2y(x+1)=5$

Solve this pair of simultaneous equations: $$\begin{cases} (x+y)^2+3y^{2}&\!\!\!\!\!=7, \\[2pt] x+2y\,(x+1)&\!\!\!\!\!=5. \end{cases} $$ I tried expanding the equations and differencing them, ...
0
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1answer
30 views

Quadratic form on Vector Bundle

A quadratic form of a vector space $V$ over a field $\mathbb{F}$ is a bilinear symmetric map $V\times V \rightarrow F$. How does one define a quadratic form over a vector bundle.
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1answer
49 views

Prove these quadractic forms are equivalent over $\mathbb{Z_5}$

Consider the following quadractic forms, defined in the field $\mathbb{Z_5}$, $$q(x, y, z, t) = 2y^2 + z^2 + 2t^2 + 4xy + 2xt + 4yt$$ $$q_0(x, y, z, t) = x^2 + y^2 + z^2 + dt^2$$ Prove they are ...
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0answers
25 views

Question on positive definiteness of non homogeneous quadratic form

I'm having trouble understanding a proposition from a semidefinite-programming textbook. It goes as follows: Let $Q$ be a quadratic function of $x \in \mathbb{R}^n$ given by $$Q(x) = ...
3
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0answers
50 views

quadratic form from nxn matrices to reals ( Tr(A^2) ). I need to find it's signature and rank.

Firstly prove $Tr(A^2)$ defines a quadratic form from the space of $n \times n$ matrices to R. I think you just have to show that $Tr(A B)$ is a bilinear form which seems too easy to be correct or I'm ...
2
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1answer
62 views

Binary quadratic forms - Equivalence and repressentation of integers

If $f,g$ are two binary quadratic forms, $f$ and $g$ are equivalent, if there is an integer matrix $M$ with determinant $\pm1 $ such that $G=M^T F M$ where $F,G$ are the matrizes that define $f,g$. It ...
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0answers
61 views

Integrate the ratio of quadratic forms

Please, help me to solve the folowing problem. Given two positive-definite $n$-dimensional matrices $A$ and $B$, need to integrate its ratio over unit ball: ...
2
votes
4answers
120 views

Find $\frac{a^3}{a^6 + 1}$ given a is a root of a quadratic equation

My question is: If a is a root of the equation $x^2 - 3x + 1 = 0$, then find the value of $\frac{a^3}{a^6 + 1}$. So, I figured we can use the Sridharacharya ...
0
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1answer
349 views

Quadratic form in canonical form

Reduce the quadratic form $q(x,y) = 6xy$ using the orthogonal reduction (i.e, find a orthogonal basis such that the matrix of the bilinear form is diagonal and $a_{ii} = 0$ or $a_{ii} = ^+_-1$) What ...
3
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1answer
31 views

equivalence of quadratic forms

Given two Hermitian positive semidefinite matrices $A$ and $B$, under what conditions on these matrices will $x^H A x = x^H B x$ for all vectors $x$? Clearly, we have equivalence when $A=B$, but I ...