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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2
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0answers
59 views

Classifing Second Degree Curves/Surfaces

I have got myself into a pickle with the following question: Classify the following (ellipse, hyperbola, ellipsoid etc) $x^2 + y^2 + 2z^2 + 2xz - 2y + 2z + 2 =0$ Now, I have written a symmetric ...
1
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1answer
55 views

Every 4-dimensional isotropic quadratic space of determinant 1 is hyperbolic

There is this claim in Scharlau's "Quadratic and Hermitian forms", Every 4-dimensional isotropic quadratic space of determinant 1 is hyperbolic. How can we prove it? I know that any ...
0
votes
3answers
127 views

What integers can be represented by the quadratic form $4x^2 - 3y^2 - z^2$?

Actually, I need to find if $4x^2 - 3y^2 - z^2 = 12$ is solvable. But I somehow feel that applying theory of integer representation by quadratic forms in three variables would yield quicker results... ...
-1
votes
1answer
78 views

Algorithm for finding full representatives of the orbit space of imaginary quadratic numbers of discriminant $D$ under the modular group

Let $\Gamma = SL_2(\mathbb{Z})$. Let $\mathcal{H} = \{z \in \mathbb{C}\ |\ Im(z) > 0\}$ be the upper half plane of complex numbers. Let $\sigma = \left( \begin{array}{ccc} p & q \\ r & s ...
1
vote
5answers
278 views

Algorithm for determining whether two imaginary quadratic numbers are equivalent under a modular transformation

Let $\Gamma = SL_2(\mathbb{Z})$. Let $\mathcal{H} = \{z \in \mathbb{C}\ |\ \mathcal{Im}(z) > 0\}$ be the upper half complex plane. Let $\sigma = \left( \begin{array}{ccc} p & q \\ r & s ...
0
votes
1answer
59 views

if f(x) is the polynomial (coeff of leadin term is unity) in 'x' of least degree such that f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1, then f(0)=?

If $f(x)$ is the polynomial (coefficient of leading term is unity) in 'x' of least degree such that $f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1$ Then $f(0)= ?$
4
votes
2answers
171 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 ...
1
vote
1answer
102 views

Elementary properties of integral binary quadratic forms

Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. $D = b^2 - 4ac$ is called the discriminant of $f$. We say $f$ is positive definite if $a \gt 0$ and $D \lt 0$(cf. this ...
1
vote
0answers
113 views

How to compute the class group of an order of a quadratic number field

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, i.e. the subring of $K$ which is a free $\mathbb{Z}$-module of rank $2$.Let $D$ be its discriminant. We use the notation and the result ...
1
vote
2answers
96 views

independent chi squares mean independent non central chi square?

Let $Y$ be a multivariate normal random vector with covariance $\Sigma$. Let $A_0,A_1$ be matrices such that $$A_0\Sigma A_1=0.$$ It is known that in this case $Y'A_0Y$ and $Y'A_1Y$ are independent ...
0
votes
2answers
591 views

Conditions for a real binary quadratic form to be positive definite

Since this question was heavily downvoted, I would like to change the presentation of the question as follows. I hope those of you who downvoted this question would be satisfied with the change. In ...
1
vote
0answers
120 views

Please help in solving $ax^2 + bxy + cx + dy + e$ = 0

Sometime back when trying to work out how to solve $ax^2 - by^2 + cx - dy + e = 0$ I learned that the way to solve such forms is to 'square the terms' and give it the form $A^2 - B^2 - E = 0$, $A = ax ...
1
vote
1answer
24 views

Explicit Isomorphism between $O(3,3)$ and $GL(4, \mathbb{R})$

I have seen it stated that $O(3,3) \cong GL(4, \mathbb{R})$, but I have never seen the isomorphism explicitly defined. Does anyone know what the isomorphism is or where I might be able to find it? ...
0
votes
2answers
42 views

From the quadtratic form of a matrix to its symmetric matrix

I am trying to solve this quadratic form: $$ f(x_1 , x_2 , x_3) = x_1^2 + x_2^2 + 5x_3^2 -2x_1x_2 + 6x_1x_2+ 3x_2x_3$$ I know that the quadratic form is defined as: $$f(x)= x^t Qx$$ However my ...
-1
votes
1answer
166 views

Relation between an integer represented by a binary quadratic form and a certain Dirichlet character defined by Jacobi symbol

Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $f$. It's easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). If ...
2
votes
1answer
217 views

A condition for an odd prime to be represented by a binary quadratic form of a given discriminant

Let $f = ax^2 + bxy + cy^2$ be an integral binary quadratic form. We say $D = b^2 - 4ac$ is the discriminant of $f$. If $D < 0$ and $a > 0$, we say $f$ is positive definite. It is easy to see ...
0
votes
1answer
263 views

how to simplify a general plane conic section's equation by linear algebra?

When encountering a general plane conic section a11x^2+a12xy+a22y^2+b1x+b2y+c=0, i can write it in matrix form as a quadratic form of the vector [x,y,1]. by what then? what should be done to reach the ...
1
vote
0answers
42 views

System of symmetric quadratic equations

Suppose $A_1, A_2, \ldots A_k$ are real symmetric (but possibly singular or indefinite) matrices. I want to know whether the system of quadratic equations $$v^T A_i v =0 $$ has a nontrivial solution ...
0
votes
1answer
41 views

Derivative of quadratic form of matrix in terms of the matrix elements?

Suppose I have $b^tAc$ and I try to get the derivative in terms of $A$. How could What is the matrix notational result? I believe the answer is $bc^t$, isn't it ?
0
votes
1answer
77 views

What is the significance of using variables h and k in vector form?

I'm just curious what the historical significance of using h,k in vector form are. It's very likely that the answer is, there is no significance just like there is no significance to using x,y,z. ...
5
votes
2answers
153 views

Why are quadratic forms so special and why not investigate in higher forms?

Ok, this is a soft question. If $K$ is a field of characteristic different from $2$, one can use the polarization identity to get a one-to-one correspondence between homogeneous polynomials of ...
2
votes
3answers
80 views

What finite fields are quadratically closed?

A field is quadratically closed if each of its elements is a square. The field $\mathbb{F}_2$ with two elelemts is obviously quadratically closed. However, testing some more finite fields on this ...
2
votes
3answers
53 views

Number of values of x

$$a\dfrac{(x-b)(x-c)}{(a-b)(a-c)}+b\dfrac{(x-c)(x-a)}{(b-c)(b-a)}+c\dfrac{(x-a)(x-b)}{(c-a)(c-b)}=x$$ How many values of $x$ satisfy this equation? It is clear that x=a, x=b, x=c do satisfy the ...
0
votes
1answer
61 views

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$.

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and either of the following is true: $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$. We see that if $(a,b,c)$ is a solution, then so is ...
0
votes
1answer
104 views

What is the geometric interpretation of quadratic forms?

I am trying to make sense of the following condition: Let $w_1, \dots, w_m \in \mathbb{C}^d$ with $\|w_i\| \le 1$ and $\sum_{i = 1}^m \, |\langle u, w_i \rangle |^2 = n$ for some $n \in \mathbb{R}$ ...
1
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3answers
65 views

What restrictions are on th sum of two fourth powers?

I've got an equation of the form $$ a^4+1=2b. \qquad(\star) $$ By well-known results regarding the sum of two squares, $b$ must be the sum of two squares. But does $(\star)$ force any other ...
0
votes
2answers
246 views

How to solve an equation of the form $ax^2 - by^2 + cx - dy + e =0$?

I am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions. Thanks!
0
votes
1answer
83 views

bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
-1
votes
1answer
80 views

The inverse class of the class represented by a primitive binary quadratic form of discriminant $D$

We use the definitions of this question. Is the following proposition true? If yes, how do we prove it? Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ ...
0
votes
1answer
38 views

Prove: matrix of the quadratic form is adj(A)

Suppose $A=\left(a_{i j}\right)_{n\times n}$ is an invertible real symmetric matrix. Prove: matrix of the quadratic form \begin{align}f\left(x_1,\text{...},x_n\right)=\left| \begin{array}{cccc} 0 ...
-1
votes
1answer
93 views

Standardize a Quadratic Form

standardize Quadratic Form $$8x_1x_4+2x_3x_4+2x_2x_3+8x_2x_4$$ how to do? what's the simplest method. How to choose the first linear replacement, matrix or do ...
1
vote
1answer
201 views

Multivariable local maximum proof

Suppose we have a twice differentiable function $f: \mathbb{R} ^n \to \mathbb{R}$, a point ${\bf x^0} = (x_1 ^0 , \ldots , x_n ^0)$ and we know that $\nabla f({\bf x}^0) = 0$ $({\bf x - x^0})H({\bf ...
1
vote
0answers
97 views

About quadratic form and its discriminant

There are 3 parts of the problem. Let d be a perfect square, possibly 0. Show that there is a quadratic form $ax^2+bxy+cy^2=0$ of discriminant d for which a=0. Let a,b,c be integers with $a\ne0$. ...
1
vote
0answers
61 views

Witt index and Pfister neighbours

Suppose $\phi$ be a 8-dimensional quadratic form with trivial discriminant over a field $F$ of characteristic not 2. Assume that there is 3-fold Pfister form $<<a,b,c>>$ such that ...
1
vote
4answers
767 views

Real world examples of quadratic and/or finding roots of a quadratic?

Anyone ever come across a good situation where a) a situation is modeled by a quadratic equation $y=ax^2+bx+c$ and/or b) you've even needed to find where $y=0$ (roots, $x$-intercept, etc)
0
votes
2answers
52 views

Get x and y in quadrat equations system

I need help in solving following system of quadratic equations : $$ 2x^2+y^2=4$$ $$2xy-2x=-5$$ I used every known me equations solving methods, but no was helpful for me... Can you help me by giving ...
1
vote
1answer
43 views

Quadratic Form $f\left(x_1,x_2,\cdots ,x_n\right)=\sum _{i=1}^m \left(a_{i 1}x_1+\cdots +a_{i n}x_n\right)^2$

Quadratic Form $f\left(x_1,x_2,\cdots ,x_n\right)=\sum _{i=1}^m \left(a_{i 1}x_1+\cdots +a_{i n}x_n\right)^2$, i) write the corresponding matrix; ii) when $a_{\text{ij}}$ are all real numbers, gives ...
0
votes
2answers
71 views

Integer roots of $x^2+y^2 = 25$ and $x^3+y^4=145$

I'm trying to find the solution of $x^2+y^2=25$ and $x^3+y^4=145$. I tried doing substitution, which leads me to: $y^2=25-x^2$ substituted to $x^3+y^4=145$ $x^3+(25-x^2)^2=145$ ...
1
vote
3answers
155 views

A problem about the quadratic form $x^TAx=0$.

$A$ is an $m\times m$ real matrix in $\mathbb{R}^{m\times m}$. If $x^TAx=0\ \forall\ x\in\mathbb{R}^m$, can we conclude that $A=0_{m\times m}$? Why? Note: $x^T$ is the transpose of $x$. $0_{m\times ...
0
votes
1answer
94 views

Extremum of a multidimensional quadratic function

I have the following function: $$ g(h) = h'\Sigma\Sigma'h-h'm-r, $$ where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
2
votes
1answer
92 views

A problem about the discrete logarithm

suppose there are a multiplicative cyclic group $F_p^*(p \;is\;big\; prime)$, and $G=\langle g \rangle(g \;is\; a\; generator)$ is a subgroup of it and $G$'s order is $q(q\;is\;big\;prime \;and ...
1
vote
1answer
312 views

How high is a baseball after 5 seconds of being thrown off of a 175 foot building? And how many seconds does it take to hit the ground? [closed]

A person standing close to the edge on the top of a 175-foot building throws a baseball vertically upward. The quadratic equation $$h = -16 t^2 + 160 t + 175$$ models the ball's height, $h$ , above ...
1
vote
4answers
408 views

How can I find integer values for which a given expression gives a perfect square?

Find the integer values for which $x^2+19x+92$ is a perfect square. Also, How to proceed if you have to find values ( not necessarily integer)?
4
votes
0answers
218 views

Proving the max of a quadratic form ${\mathbf x}^T\mathbf A \mathbf x$ can be attained when $x$ is from $n$-dimensional hypercube

updated: Maybe my original question is somewhat misleading. I rewrite some of the post. This is some research problem I'm working on. I have an $n\times n$ symmetric positive-definite matrix ...
0
votes
1answer
283 views

Ratio of dependent chi squared random variables

Suppose that $X=v'A_1v$ and $Y=v'A_2v$, where $A_i$ are symmetric matrices and $v$ a multivariate normal vector with covariance $V$, are chi squared distributed each with its own degrees of freedom. ...
2
votes
1answer
81 views

finding zeroes of a quadratic form

Let $a,b\in\mathbb Z$ be squarefree with $a>0$. Suppose that I know that there exist $(0,0,0)\neq (x,y,z) \in \mathbb Z^3$ s.t. $x^2-by^2-az^2=0$. Is there any known algorithm to find any such a ...
2
votes
1answer
213 views

Graphing quadratic form, which eigenvalue should be chosen first?

I just graph a quadratic function, $-4x^2_1+4x_1x_2-7x_2^2=-8$, by: Find the eigenvalues of the function above, which are $\lambda_1=-8$ and $\lambda_2=-3$ Use the eigenvalues to make a new ...
2
votes
1answer
73 views

Are $Q_1(x,y)=xy$ and $Q_2(x,y)=x^2+y^2$ equivalent forms?

This is an old qual problem at my school. I've never dealt with quadratic forms, so I'm not sure if I've done this right. Two quadratic forms $Q_1(x,y)$ and $Q_2(x',y')$ are said to be equivalent ...
1
vote
1answer
43 views

Showing a particular function on the space of polynomials w/ degree $ \le 2$ is a quadratic form, and computing signature

I've been a long way from linear algebra but I have to go back to it for an exam, and I've found myself stuck on the following question. Define $Q$ on the space of all polynomials with degree at most ...
4
votes
2answers
446 views

Clifford Algebras

What would be the best source to learn Clifford Algebras from? Anything online would suffice or any textual sources for that matter.. I'm interested in doing a project in the subject, but I'm not ...