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Questions tagged [quadratic-forms]

A quadratic form is a homogeneous polynomial of degree two (in any number of variables), for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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Assumption $d>2$ on Proposition 2.12 from Knapp's Elliptic Curves

I'm going through Knapp's book on elliptic curves and I got stuck in a minor detail. This is a part of the proof of Proposition 2.12: I could understand everything except for this little detail: ...
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Find parameter a for which..

Find parameter $a$ for which $$\frac{ax^2+3x-4}{a+3x-4x^2}$$ takes all real values for $x \in \mathbb{R}$ I have equated the function to a real value, say, k which gets me a quadratic in x. I have ...
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Finding representative matrix of quadratic form $ q(x) = x_1x_2 + x_2x_3 + x_1x_3$ with respect to a basis

I'm slightly confused about this question: we have a basis of $R^3$ $B = ${$e_1,e_2, e_3$}, where $x = x_1e_1 + x_2e_2 + x_3e_3$. We need to give the representative matrix of $q$ with respect to ...
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Gradient of an $n$-variate quadratic form

Given function $$g(u) = (Au)^T(Au)$$ where $u \in \mathbb{R}^{n}$ and $A$ is a matrix of dimension $n \times n$, find the gradient $\nabla g(u)$. I tried to expand everything and got that $$\frac{\...
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How do i derive the equation of a Bezier curve?

I have encounted a problem so i am trying to create and plot a Bezier curve and i have four control points. I have to link the application of polynomials and Pascals triangle within the answer. Now i ...
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How to find the projection along the following vector subspace?

I am given with the inner product, $$\phi(a,b) = a_1b_1+a_2b_3 + a_3b_2$$ where $a=(a_1,a_2,a_3)\text{ and } b= (b_1,b_2,b_3)\in \mathbb{R}^{3}.$ Consider the vector space $F = \text{span}(1,1,1).$ ...
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Is it easy to check if two matrices define the same quadratic form over $\mathbb{Z}$?

Given two $\ell \times \ell$ symmetric matrices, is there an easy way to check if they define the same quadratic form over $\mathbb{Z}$ (up to a change of basis)? In particular, among other examples,...
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Quadratic form vanishing at certain points

Let $A\in\mathbb{R}^{d\times d}$ be a symmetric matrix, and $X_1,\dots, X_n\in \mathbb{R}^d$ be vectors with $n>d$ (if more convenient, one can assume ${\rm span}(X_1,\dots,X_n)=\mathbb{R}^d$. ...
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ternary quadratic forms over polynomial rings

The following theorem by Legendre is well-known: The integral ternary quadratic form $f(x,y,z) = ax^2 + by^2 +cz^2 \in {\bf Z}[x]$ represents $0$ non-trivially if and only if $a$, $b$, and $c$ do not ...
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Solve $x \circ a + x \circ (Bx) + c = 0$ for $x$?

Is there a solution to $x \circ a + x \circ (Bx) + c = 0$ for $x$, where $B$ is an $N \times N$ matrix, $x$, $a$ and $c$ are $N \times 1$ column vectors, and $\circ$ is the Hadamard product (element-...
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Question about “Primes of the form $x^2+ny^2$”: Proper ideals are invertible

I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma: Lemma 7.5: Let $...
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21 views

Quadratic form, Lagrange method

I can't transform quadratic form to canonical using Lagrange method. I know how it works, but still can't. Form is $x^2 +4xy +8xz -3y^2 + 5z^2$. I have got some variants, but there is always something ...
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Representation by a quadratic binary form

For $m$ a non-zero integer and discriminant $d=b^2-4ac$ congruent to either 0 or 1 modulo 4, show that m is properly represented by some binary quadratic form $f(x)=ax^2+bxy+cy^2$ if and only if the ...
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Сanonical form of a quadratic form by Lagrange's method

This is just an eg. to understand how to solve it. There is a quadratic form as below we need to bring to canonical form using Lagrange's method and find the coordinate transformation. Then find ...
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Derivative of a quadratic form — how to derive it?

I want to know how $$\frac{\delta(x^TAx)}{\delta(x)}=2Ax$$ I think here's what happens (please correct me where wrong): (by: the rule for matrix derivative) $$ \frac{\delta(x^TAx)}{\delta(w)}=x^T(A^...
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Positive-definite over $\mathbb Q$ form is positive definite over $\mathbb R$?

I was reading P. Etingof's "Introduction to the representation theory" when I found this problem (and I've trouble with it): we have a quadratic form $Q(x) = \sum x_i^2 - \frac{1}{2}\sum b_{ij}x_ix_j$,...
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Signature of a bilinear form in the space of $2 \times 2$ Hermitian matrices

We have a bilinear form, $$\langle A, A' \rangle = \det(A+A') - \det A - \det A'$$ on the real vector space of Hermitian $2\times 2$ matrices. If I have a basis of the space, I would calculate ...
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Make Ising-like quadratic form positive definite

Consider a quadratic form $Q_a:\mathbb{C}^N\rightarrow\mathbb{C}:s\mapsto Q_a(s)=(a+it)\sum_{i=1}^N s_i^2-\beta J\sum_{\langle i,j\rangle}s_is_j$. The idea here is that at every point $i$ of a lattice ...
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Hessian and quadratic form

I can't understand what my professor meant in our Calc III book. In a demostration, he wrote something like "we are going to refer to the hessian ($Hf_{P}$) and its associated quadratic form ($Q_{P}(V)...
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Find a basis $B$ such that the matrix representation of the quadratic form $q$ will be a diagonal matrix. Why is this solution correct?

We are given the quadratic form $q((x,y,z)) = (x-y+2z)^2+8y^2-2(z-2y)^2$. Therefore if $x' = x-y+2z, y' = 8y^2, z' = z-2y$ we have $q((x,y,z))=x'^2+8y'^2-2z'^2$. In order to find the basis $B$ where ...
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how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$

I have a problem with the following question: how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$. How I tried to solve this problem. Here we have ...
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$k\sum v_i v_i^T-\big(\sum v_i\big)\big(\sum v_i^T\big)\succeq 0$

My professor claimed that $$k\sum_{i=1}^k v_i v_i^T-\Big(\sum_{i=1}^k v_i\Big)\Big(\sum_{i=1}^k v_i^T\Big)\succeq 0,$$ holds for any family of vectors $\{v_1,\dots,v_k\}$, and can be shown using the ...
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Find the points of the surface defined by $-x_1^2 + x_2^2 - x_3^2 + 10x_1x_3 = 1$ closest to the origin

For the surface $$\{ (x_1, x_2, x_3) \in \mathbb R^3 \mid -x_1^2+x_2^2-x_3^2 + 10x_1x_3=1 \}$$ I want to find the points on it closest to the origin. I know how to do a substantial part of this. ...
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Why does this inequality with quadratic forms hold?

Let $A, B$ be $n \times n$ positive definite matrices, and let $x$ be an $n$-vector. Why does the following hold? $$ x^{T}(A^{-1/2} (A^{1/2} B^{-1} A^{1/2})^{1/2} A^{-1/2})x \leq (x^{T}A^{-1}x)^{1/2}...
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Cochran's Theorem

I am preparing a work on Cochran's theorem and I had two questions : First question : Is there a link between these two statements and to what extent? Would it be redundant to prove each of them ...
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Quadratic Forms and Dual Space

Suppose we have $V$, a finite $k$-vector space, where k is a field with $char(k) \neq 2$. And we have a quadratic form $Q$ on $V$. If $Q$ is non-degenerate then the map $q_v:V \to V^*$ sending $v \in ...
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Prove of quadratic Optimal control

Given system without control $\dot{x}=ax.$ Given definiton of function in interval $[t,T]$ as $$J(t)=\frac{1}{2}S(T)x^2(T)+\frac{1}{2} \int_{t}^{T}qx^2(\tau)d\tau.$$ Its Lyapunov equation is: $$-\...
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What is the point of finding the standard form of a quadratic form?

In a problem sheet for my linear algebra course I was asked to find the standard form of the quadratic form $Q=8x_1^2 + 2x_2^2 +3x_3^2 +8x_2x_3$. Following the steps in the lecture notes I arrived ...
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Issue understanding quadratic forms

In my linear algebra course my professor asked us this question in the problem sheet. Consider the quadratic form $Q = 8x_1^2+2x_2^2+3x_3^3+8x_2x_3$. Find $\alpha$, $\beta$ and $\gamma$, such that $\...
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Prove that $\sum_{i=1}^{n}\frac{1}{OA_i^2}$ is constant for ellipsoid

I have the following problem. Problem. Ellipsoid $\mathcal{E}$ with center $O$ in euclidean space $\mathbb{R}^n$ is defined by equation $\dfrac{x_1^2}{a_1^2}+\dfrac{x_2^2}{a_2^2}+\ldots+\dfrac{x_n^...
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Let $q$ be a quadratic form on the finite vector space $V$ over $\mathbb{F}_2$.

And let $q$ be such that the bilinear form $B: (x, y) \mapsto q(x+y)+q(x)+q(y)$ is nondegenerate. Let $$A(q)=\frac{\sum_{x\in V}(-1)^{q(x)}}{\sqrt{|V|}}\in \mathbb{R}.$$ I need prove that $A(q)$ is $1$...
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Choice of Polarization of an algebraic form as a optimization tool in the context of quadratic forms instead of Derivative test

Formally, an inner product space is a vector space V over the field F together with an inner product, i.e., with a map $\langle \cdot ,\cdot \rangle :V\times V\to F$ that satisfies the following ...
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Classical $\sum_{1\leq i\leq n} x_i^2$ are group forms for $n=1,2,4,8$.

Let $F$ be a field. Consider quadratic forms $f=\sum_{1\leq i\leq n}x_i^2$ with $n=1,2,4,8$. $f$ is a group form if $\{d\in F^\star|\exists x\in F^n, f(x)=d\}$ is a group. For $F=Q$, it follows from ...
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185 views

Proving an identity of quadratic form

Let $n$ be even and $x_1,x_2,⋯,x_n$ be reals. Show that$$\sum_{1\le i<j\le n}\min(|i-j|,n-|i-j|)x_ix_j\\=\sum_{j=1}^{\frac n2}(x_j+x_{j+1}+⋯+x_{j+\frac n2-1})(x_{j+\frac n2}+x_{j+\frac n2+1}+⋯+x_{j+...
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Isomorphic elements in $H^i(k;\mu_2)$

An element $\alpha \in H^i(k;\mu_2)$ is called a pure symbol in case it has the form $\alpha = (a_1) \cup\dots\cup (a_i)$, $(a_i) \in H^1(k;\mu_2)$. It is known that for every pure symbol $\alpha $ ...
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Binary Quadratic Forms of Discriminant -3

The following is a question in my textbook: Show that any positive definite binary quadratic form of discriminant $-3$ is equivalent to $f(x, y) = x^2 + xy + y^2$. Show that a positive integer $n$ ...
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1answer
69 views

Decomposing bivariate quadratic form into sum of two squares

I would like to decompose $$ax_1^2 + bx_2^2 + 2cx_1x_2$$ into two expressions, each involving only one variable. I'm trying to use a transform like $x_1 = x_+ + x_-$ and $x_2 = x_+ - x_-$ to ...
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Determine the set of integers that are represented by the binary quadratic form (1,0,-1) [duplicate]

I need help with finding the set of integers represented by the form (1,0,-1). This is essentially f(x,y) = x^2 - y^2 which can be factorised into (x + y)(x - y) and the determinant is d = 4 > 0.
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Proof of derivative of $x^TBx$ using the product rule

I'm trying to prove that when $f(x) =x^TBx$, then $f'(x) = (B + B^T)x$. I haven't found this formula online but going through the calculations using index notation this is what I came up with. This ...
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$SL_2 (\Bbb R) × SL_2 (\Bbb R)/ ± (I_2 , I_2 ) → (SO_{2,2})^\circ$

I went through this problem in Lie groups: i) Prove that $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ is a linear Lie group. I identified $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ with $\{\begin{pmatrix} A & 0 \\ 0 &...
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On the signature of a quadratic form

Prove that the determinant in $M_2(\Bbb R)$ is a quadratic form of signature $(2,2)$. I found the first part: the symmetric bilinear form $$B(M,N)=\frac{1}{2} ( \det(M+N) - \det (M) -\det (N) )$$ ...
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How to reduce quadratic form using orthogonal transformation

I wanted to reduce following quadratic form $f(x,y,z)=3x^2+3y^2+3z^2-xy-yz$ I found its symmetric metric but its eigenvalues too weird to start to for basis that convert to Eigen basis. I am new to ...
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Expressing quadratic form of normal variables in terms of chi-squared variables

On Wikpedia and in the references therein [1,2], it is stated that any quadratic form of normally distributed random variables can be expressed as the sum of many independent non-central chi-squared ...
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42 views

Is there an easy way to tell if a set of quadratic constraints are solvable

I am trying to find a matrix whose null space $N\in\mathbb{R}^{9 \times n}$ does not intersect with the column space of the matrix $$ M(R) = \begin{bmatrix} 0 & -r_3 & r_2 \\ r_3 & 0 &...
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smooth projective quadric $Q_{n,m}$ in $\mathbb{P}_n(\mathbb{R})$ with planarity $m$

How to show that smooth projective quadric $Q_{n,m}$ in $\mathbb{P}_n(\mathbb{R})$ with planarity $m$, dimension $n$ and $i=p-q$ (where $(p,q)$ is signature of the quadratic form) are related by $2m+i=...
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Form Class group - Special cases

I am trying to find special cases for when the form class group will have a predictable structure. I am specifically interested in the case of prime discriminants or relating the structure for non-...
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20 views

Structure of class group of quadratic forms negative discriminant

I am trying to work out the structure of the class groups for discriminants of some simple forms. I am interested in the cases where the discriminant is a prime number and where it is of the form $d =...
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Representation of Generalized Quadratic Form of Random Variables

I am interested in finding/understanding a "good" $L^2$-orthogonal (i.e. uncorrelated) decomposition of a matrix-valued quadratic form. The setup is as follows: Given a random matrix $X$ (tall, of ...
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1answer
27 views

Quadratic Forms Orthogonal Diagonalization Existence

Why does one assume that the eigenbasis for a quadratic form is orthogonal, hence orthogonal diagonalization. I understand that for hermitian and unitary maps one can show by spectral theorem an ...
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Deterministic algorithm for class group of Binary quadratic forms

I want to find the class group of a given negative discriminant. I know of Shanks method but this is not deterministic. I can of course brute force the problem by finding the class group and then ...