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

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3answers
65 views

Express a quadratic form in three variable in the format $x^tAx$ using a substitution $x=Py$

I was asked to determine if a quadratic form is positive definite. To do so I must convert in the format $x^tAx$ using a substitution x=Py. So that "it can be written in diagonal form". $$Q(x,y,z) = ...
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3answers
74 views

Finding the matrix of a quadratic form

I want to find the matrix of quadratic form $Q= \sum^p_{i=1} (y_i - \bar y)^2$. Please help me finding it. For example I have found the quadratic form matrix for $Q= p\bar y^2$ as follows: $$Q= ...
0
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1answer
38 views

Compute the dimension of the space of quadratic forms

We were asked the following: "Compute the dimension of the space of quadratic forms on $V=\mathbb{R^2}.$ Compute also the dimension of the space of symetric forms on $\mathbb{R^2}$, $S^2\mathbb{R^2}$....
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0answers
91 views

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

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
0
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0answers
25 views

Quartic polynomial in ten variables

I have a quartic form, i.e. a homogeneous 4-th degree polynomial, in ten real variables and the inequality: $f(x_1,\ldots, x_{10}) \geq c$, for some $c>0$, which I believe that geometrically ...
3
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0answers
18 views

$q_I$ primitive as a quadratic form? [closed]

Let $I \subset \mathcal{O}_k$ be an ideal, $N(I) = [\mathcal{O}_K : I] = |\mathcal{O}_K/I|$. Define $q_I$ be $q_I(x) = N_{K/\mathbb{Q}}(x)/N(I)$. Is $q_I$ primitive as a quadratic form?
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23 views

Optimizing the sum of powers of positive quadratic functions

In my research I have come across the following optimization problem. \begin{equation} \begin{array}{c} maximize \hspace{1cm} \sum \limits_{n=1}^{N}\left(\mathbf{x}^{T}A_n \mathbf{x}\right)^{k}\\ s....
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1answer
26 views

Matrix of quadratic form (in Serre's general notion)?

I am currently reading Serre's book on arithmetic. In chapter four (page 27) he defines a general notion of the quadratic form as: Let $V$ be a module of a commutative ring $A$. A function $Q$ is ...
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1answer
30 views

Solution of quadratic diophantine equations

Is there any algorithm so that solution to the following equation can be found? $(x+a)^2-y^2=c$ where $c$ and $a$ is a constant. It is similar to Pells eqution with a variation where $D=1$. I am new ...
0
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1answer
34 views

Is the following inequality correct?

I'm trying to understand whether the following inequality is correct. Let $Y,X$ be random variables and $f(X)$, $n\times 1$-dimensional function of $X$. It is claimed that $$\begin{aligned} & a'(A'...
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0answers
44 views

Order of the sum of elements of the inverse of a matrix

For each $T$, let $A_T$ be a $T\times T$ matrix of real numbers. let $e_T$ be the $T\times 1$ vector of ones. Assume that the sum of all entries of the matrix $A_T$ divided by $T^2$ is limited as $T$ ...
0
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0answers
77 views

Vieta Jumping and Hurwitz 1907

Today I proved finiteness for the problem here: Is it true that $f(x,y)=\dfrac{x^2+y^2}{xy-t}$ has only finitely many distinct integer values with $x,y$ positive integers? namely: IF we have ...
0
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1answer
28 views

Symmetric part of A contributes to quadratic form

In my statistics note, when it talks about quadratic forms, it goes on saying: "$x^tAx=\frac12x^t(A+A^t)x$ implies that only the symmetric part of A contributes to the quadratic form." I am having ...
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0answers
25 views

How to write the symmetric Hessian matrix for a log function?

Say f(x,y,z) = $y*ln(cos(z)+x^2)$ How would I write this as a Hessian matrix? Would this be the right step I need to take in order to calculate the second-order Taylor polynomial for the function?
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0answers
11 views

Representation of integers by sum of squares with linear constraints of a special form

I would like to know what integers $d$ can be written as a sum $d=\sum_{i=1}^N\sum_{j=1}^M a_{ij}^2 $ with $a_{ij} \in \mathbb{Z}$ and where the row and column sums of $a_{ij}$ are fixed $\sum_{i=1}^N ...
1
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0answers
77 views

Linear reformulation or approximation of a quadratic inequality set

Based on the useful comments I reformulated my problem - hopefully it's more clear now. Let $A,B \in \mathbb{R}^{d \times d}$ be symmetric positive-semidefinite matrices, $x \in \mathbb{R}^d$ and $||...
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1answer
30 views

Lagrange Method of Quadratic Form the a Billinear Form

In the following question I have to present the bilinear form as sum of squares with Lagrange method. $$q(x_1,x_2,x_3,x_4)=2x_1x_4-6x_2x_3$$ However I don't know how I can do it here since none of ...
2
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2answers
79 views

Expectation of Univariate Quadratic Form under Multivariate Gaussian

Is there an obvious trick I am missing for solving the following integral: $$ \int_x P(y|x) W(x) (-x^TMx+2x^Tm -c)dx$$ Distributions are Gaussians and $M$ is symmetric. I know how to do the ...
1
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1answer
36 views

How many different values can $(x^2 + y^2, x^2 + 2y^2 )$ have mod 4?

It is known that if a prime number $p = x^2 + y^2 $ is equivalent to $p \equiv 1 \mod 4$. This is Fermat's theorem on the sum of two squares. My question is about the value of two simultaneous ...
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1answer
28 views

Are all “forms” linear maps from vector spaces to fields?

It seems that whenever we call something a "form": quadratic form, linear form, bilinear form, one-form, two-form, etc. it is always a linear (or perhaps not?) map from some vector space (or elsewhere?...
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2answers
44 views

Why is the inner product of a quadratic form a quadratic form?

I was going through a derivation of the second derivative of the $\log \det X$ where $X$ is symmetric positive definite, I noticed that despite the second order approximation of log det is written as: ...
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2answers
49 views

Assume this equation has distinct roots. Prove $k = -1/2$ without using Vieta's formulas.

Given $(1-2k)x^2 - (3k+4)x + 2 = 0$ for some $k \in \mathbb{R}\setminus\{1/2\}$, suppose $x_1$ and $x_2$ are distinct roots of the equation such that $x_1 x_2 = 1$. Without using Vieta's formulas, ...
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1answer
72 views

When are we allowed to match coefficients?

Related to this answer: Find k in $(1−2k)x^2−(3k+4)x+2=0$ given facts about the roots. In the partial fraction decomposition of $\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$, we have: $0x + ...
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2answers
84 views

Find k in $(1-2k)x^2 - (3k+4)x + 2 = 0$ given facts about the roots.

The exact instruction in my book is: A quadratic equation $(1-2k)x^2 - (3k+4)x + 2 = 0$ is given. Find the value of k for each of the following conditions. (I got 46a and 46b) (c) A ...
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0answers
25 views

Normal forms of a quadratic form of two variables.

If we are given a form $Q(x,y) = ax^2 + 2bxy +cy^2$, then, using a rotation (or a linear change in coordinates) we may eliminate the $xy$ cross term to obtain the following form: $Q(X,Y) = AX^2 + CY^2$...
0
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1answer
53 views

Primes of special form

Are there infinitely many primes $p$ of form $$2^k+a^2=p^2<2^{k+2}$$ where $a\in\Bbb N$? Which primes are known to be of such form? An example is $16+3^2=5^2$. This is the only one I could find.
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1answer
57 views

Sum of squares as Primes Class Field Theorem statements

We know that every prime $1\bmod 4$ can be written in an unique way as $a^2+b^2$ form where $a,b\in\Bbb N$. Is there a comprehensive list of other statements of form "every prime $d\mod r$ can be ...
2
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2answers
66 views

Quadratic form has non-trivial zero?

For each of the following quadratic forms, determine whether the form has a non-trivial zero (we do not need to exhibit it): $f(x, y, z) = 2x^2 + 3y^2 - 6z^2$; $g(x, y, z) = 2x^2 + 3y^2 - ...
2
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4answers
182 views

Is the matrix $A$ positive (negative) (semi-) definite?

Given, $$A = \begin{bmatrix} 2 &-1 & -1\\ -1&2 & -1\\ -1& -1& 2 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Define the ...
0
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1answer
63 views

How many ways can a quadratic form represent a prime?

Given $a,b,c,p\in\Bbb N$ with $b^2-4ac<0$ and $p$ is a prime with $\bigg(\frac{b^2-4ac}p\bigg)=1$, how many solutions $(x,y)\in\Bbb Z^2$ are there to $$ax^2+bxy+cy^2=p?$$
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1answer
65 views

How to transform the quadratic form of an ellipse to a circle

Consider the ellipse $x^TPx\le a$. I would like to transform (the quadratic form of) this ellipse into a circle $y^T\begin{pmatrix}1&0\\0&1\end{pmatrix}y\le b$ via a coordinate transform $x=Ty$...
1
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1answer
39 views

quadratic form in hilbert space and Gram matrix

We are in Hilbert space $L^2$ we are given a subspace of dimension K as $$ V=\{ g_k,1 \le k \le K \} $$ everything that folows is defined on $V$ we define map $$ x \mapsto Q(x):= \sum_{k=1}^{K} |\...
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1answer
63 views

Quadratic form in Hilbert space associated with orthogonal projection operator

we are in Hilbert space $L^2 $ and we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
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0answers
34 views

Moments of quadratic forms

$x=(x_1,...,x_T)'$ is a $T\times1$ random vector, where $x_t, t=1,..., T$, is a stationary process with mean zero and finite fourth moments. $A$ is a $T\times T$ symmetric constant matrix. How to find ...
0
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4answers
80 views

Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.

I try to transform Transform $$f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$$ to a diagonal form. I can do it using eigenvalue, but when I directly complete the square to find its ...
3
votes
2answers
100 views

Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
0
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0answers
67 views

Simultaneously diagonalise two real quadratic forms

I would like to simultaneously diagonalise the quadratic forms $A=2x^2+3y^2+3z^2-2yz$, and $B=x^2+3y^2+3z^2+6xy+2yz-6zx$. Of course there's a theorem saying this is possible and I followed the ...
2
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1answer
71 views

How to compute the matrix $S$ in Sylvester's law of inertia

Sylvester's law of inertia states that for any symmetric matrix $A$ there exist an invertible matrix S such that, $S^T A S = D$, where $D$ is a diagonal matrix which has only entries 0, +1 and −1 ...
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2answers
23 views

Can I find a vector with entirely nonzero entries for the following quadratic form to evaluate to zero?

I have a square, symmetric matrix $M$, of size at least $2\times 2$, with diagonal entries equal to $1$ and off-diagonal entries equal to $\pm 1$. Let the entries of $M$ be such that it is indefinite. ...
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1answer
52 views

Omission in Jacobson's BAI regarding extension of isometries.

Suppose $V$ is a finite dimensional vector space over a field of characteristic $\neq 2$ equipped with a nondegenerate quadratic form $Q$. Witt's cancellation theorem says that if $U_1,U_2$ are ...
3
votes
2answers
125 views

Multivariate Gaussian integral of ratio of quadratic forms

Given two real symmetric matrices $M,S$ is there a known answer for the Gaussian integral $\int d^Nz\frac{z^TMz}{z^TSz}$ where the integration is over N-dimensional Gaussian variable $z\sim N(\vec{0},...
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3answers
228 views

reference for linear algebra books that teach reverse Hermite method for symmetric matrices

January 13, 2016: book that does this mentioned in a question today, Linear Algebra Done Wrong by Sergei Treil. He calls it non-orthogonal diagonalization of a quadratic form, calls his first method ...
2
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2answers
165 views

Find the transitional matrix that would transform this form to a diagonal form.

Let the quadratic form $F(x,y,z)$ be given as below $F(x,y,z)=2x^2+3y^2+5z^2-xy-xz-yz$ Find the transitional matrix that would transform this form to a diagonal form. I got the symmetric ...
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0answers
50 views

Minimizing a quadratic form with orthogonality constraints

Suppose $A$ is an $n$-by-$n$ symmetric matrix, and I want to find $x_{1}$ and $x_{2}$ that maximize $x_{1}^{T} A x_{1} + x_{2}^{T} A x_{2}$ subject to the constraint that $x_{i}^{T} x_{j} = \delta_{ij}...
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0answers
49 views

Find Isotropic vectors that form a basis

I have this question: let $(E,\langle,\rangle)$ an inner product space with dimension $n$ and $u$ a symmetric linear transformation and we define a quadratic form $q$ by $$\forall x\in E,\quad ...
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2answers
32 views

Solve the algebraic expression for a, b, and c of the function x

I am trying to solve for $a$, $b$, and $c$ in the expression below, but I have found that the way I tried to solve it is convoluted and did not work out. I believed that by solving for x, I would be ...
4
votes
1answer
143 views

Reducing a pair of indefinite quadratic forms to the canonical form

Assume $A, B$ being a pair of symmetric matrices over reals. Let $$ \varphi_1(x) = (x, Ax)\\ \varphi_2(x) = (x, Bx). $$ There's a well-known result that if $A > 0$ then the pair of forms can be ...
6
votes
4answers
189 views

Parametrization of $x^2+ay^2=z^k$, where $\gcd(x,y,z)=1$

$x,y,z$ be three coprime integers, $a \in \mathbb{Z}>0$ and $k$ an odd integer. How do I find all the non-trivial solutions of the diophantine equation? $$x^2+ay^2=z^k$$ Does the method which ...
9
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0answers
88 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
5
votes
2answers
196 views

How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...