Questions tagged [quadratic-forms]
Quadratic forms are homogeneous quadratic (degree two) polynomials in $n$ variables. In the cases of one, two, and three variables they are called unary, binary, and ternary. For example $\quad Q(x)=2x^2\quad $ is called unary quadratic ploynomial, $\quad Q(x,y)= 2x^2+3xy+2y^2\quad$ is called binary quadratic polynomial and $\quad Q(x,y,z)=2x^2+3y^2+z^2+7xy+5yz+9xz\quad$ is called ternary quadratic polynomial.
2,420
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Decisions and choices problem, matrix of its coefficients
I have a problem finding a matrix of coefficients for $$Q(x,y,z)=3x^2+2y^2+3z^2-2xy-2yz$$ original question is "Find the eigenvalues and the eigenvectors of the matrix of its coefficients"
I ...
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Sequence of Quadratic Form of Random Variables
I want to proof that
$\mathbb{E}\left[\frac{1}{n}\sigma({u}^{T})w_{n}w_{n}^{T} \sigma(v)\right] \overset{n\rightarrow \infty}{=} \mathbb{E}\left[\sigma({l})\sigma(s)\right]$
with l,s beeing Gaussian ...
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$ n $ quadratic forms in $ n+1 $ variables
This answer seems to imply that there is something special about a system of $ n $ quadratic forms $ q_1, \dots, q_n $ in $ n+1 $ unknowns $ x_1, \dots, x_{n+1} $. I want to understand better why this ...
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Finding every solution of $a^2+b^2+c^2=3$ in $\mathbb{Q}(i)$
Specifically, $a,b,c\in\mathbb{Q}(i)$ are complex numbers with rational parts whose squares sum to $a^2+b^2+c^2=3$. There's an answer to this question over $\mathbb{Q}$ already here but I couldn't ...
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Lower bound on quadratic form with positive definite matrix
I came accross the following lower bound and I am looking for a proof that uses only basic properties or well-established results of linear algebra (or other domains of math).
Let $M \in \mathbb{R}^{d\...
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prove the dimension of annihilation subspace of quadratic form is lower than $\min\{n-p,n-q\}$
Assume $f(x_1,x_2,\cdots,x_n)$ is a quadratic form on $\mathbb{R}^n$. The positive and negative inertia index is $p$ and $q$. It is evident that the vector $\alpha$ that satisfies $f(\alpha)=0$ form a ...
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Given $y$, is $-\left(\dfrac{x - y}{2}\right)^2$ is a quadratic residue $\pmod x$?
I am trying to find a formula for some circle packings when the following arose. I am wondering if there is a nice way to find which $y$ yield $-\left(\dfrac{x - y}{2}\right)^2$ a quadratic residue $\...
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Non-equivalent quadratic forms with same genus
in the wikipedia page about the genus of a quadratic form the wikipedia page about the genus of a quadratic form it is claimed that $Q(x,y)=x^2+82y^2$ and $P(x,y)=2x^2+41y^2$ are not equivalent (over $...
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How to extend an orthogonal linearly independent set to an orthogonal basis, with respect to a symmetric bilinear form?
Let $F$ be a field whose characteristic is not $2$. Let $X$ be an $n(<\infty)$-dimensional vector space over $F$, equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$ on $X$. Let $...
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Existence of a $q$-orthonormal basis of $\mathbb{R}^{r+s}$
For some propositions and proofs it was assumed that there exist a $q$-orthogonal (respect. $q$-orthonormal) basis of $\mathbb{R}^{r+s}\subset Cl_{r,s}$. Here $q$ is a quadratic form. Since we ...
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Number of Classes in a Genus of a Quadratic Form?
I have been studying the representation of integers by sums of squares. Consider the equation
$$x_1^2+\cdots+x_n^2=k,$$
and let $r_n(k)$ denote the number of integer solutions. One way to compute $r_n(...
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Binary program that maximizes ratio of quadratic forms
I'd like to solve the following optimization problem. Given $\mathbf a, \mathbf b \in (0, \infty)^n$, find $\mathbf x \in \{0, 1\}^n$ which maximizes
$$ f (\mathbf x) = \frac{\left( \sum\limits_{i=1}^...
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Relation between the level and the determinant of a matrix
Let $ \mathcal{D}_k$ be the set of $k \times k$ integer, positive definite matrices with even diagonal. For $A \in \mathcal{D}_k$ we define the level of $A$ as the smallest integer $n_A \in \mathbb{N}...
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Necessary and sufficient condition to write a quadratic form on a finite-dimensional real vector space as a product of two linear functionals
I have come across a tricky linear algebra problem. We want to prove that a quadratic form $q$ on a finite dimensional real vector space $V$ can be expressed as $q(v) = f_1(v)f_2(v) \iff r + |\sigma| \...
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Seeking the solution of an equation with quadratic kernel matrix as variable.
I tried to solve the following equation in matrix ${\bf A} \in \Bbb R^{N \times N}$. There are a total of $M$ known vectors.
$$\textbf{x}^{(i)}=(x_1^{(i)},x_2^{(i)},...,x_N^{(i)})^T \in \Bbb R^N,i=1,...
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Non-degenerate bilinear forms and orthogonal bases for free modules
Given a finite-dimensional free module $M$ over a (commutative) ring $R$, let $(-,-)$ be a symmetric non-degenerate $R$-bilinear map $M \times M \to R$. Will $M$ admit two $R$-module bases, $e_i$ and $...
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Equivalence of quadratic forms if they have the same signature.
While studying for linear algebra, I came across an assignment that I'm having trouble with. I don't know a bit which side to start from. Namely:
We say that two quadratic forms $a: V \to K$, $b: W \...
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Proof of $\operatorname{rank} \left( X’ V X \right) = \operatorname{rank} (X)$ [closed]
From page 39 of George Seber's A Matrix Handbook for Statisticians:
Let $V$ be a non-negative definite $n \times n$ matrix, and let $X$ be an $n \times p$ matrix. Then the following statements are ...
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Show that the action of PSL$_2(\mathbb Z)$ on a quadratic form by $g \cdot Q = Q(ax + by, cx + dy)$ preserves the set of properly represented numbers
I am trying to show that
the action of PSL$_2(\mathbb Z)$ on a quadratic form by $g \cdot Q = Q(ax + by, cx + dy)$ preserves the set of properly represented numbers, where
$$g = \begin{pmatrix}
...
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Clifford Algebra with $Q$=0
On Wikipedia it says:
Clifford algebras are closely related to exterior algebras. Indeed, if $Q$ = 0 then the Clifford algebra $Cl(V, Q)$ is just the exterior algebra ⋀$V$.
With $Q$ being a quadratic ...
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Determinant is a quadratic form
Let $H=\{A \in M_{2\times 2}(\mathbb{C}) \ | \ A = A^\dagger := \overline{A}^t \}$ be the real vector space of all hermitian $2 × 2$ complex matrices.
(a) Show that for all $A \in H$, $\text{det}(A)$ ...
user1177059
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Second order derivative of $f(x):=\frac{1}{2} ⟨x,Ax⟩$
Let $A=\left(A_{i j}\right)$ be an $n \times n$ symmetric matrix, and define the function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ as
$$
f(x):=
\frac{1}{2}
⟨x,Ax⟩
$$
Using the definition, ...
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What is wrong with my approach to finding solution of these equations?
I have to find solutions $ \mathbf {(x,y)}$ where $ \mathbf x$ and $ \mathbf y$ are real numbers for the system of equations
$\mathbf {x^2-xy+y^2=21}$
$\mathbf {x^2+2xy-8y^2=0}$
what i initially did ...
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Solve $x, y \in \mathbb{R}$ given $x(12-x)+y(16-y)=100$.
Solve the equation in $\mathbb{R}$
$$x(12-x)+y(16-y)=100$$
MY IDEAS:
\begin{aligned}
& x(12-x)+y(16-y)=100 \\
& 12 x-x^2+y^{16}-y^2=100 \\
& 12 x+16 y-x^2-y^2=100 \\
& 12 x+16 y-x^2-y^...
user1104319
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Is it possible to determine the number of "standardized" quadratic forms?
Given a quadratic form $q$ on $\mathbb R^n$, are there methods to determine the number of "standardized" quadratic forms using the Gauss's square decomposition algorithm (https://fr....
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Show that we have an induced quadratic form on quotient vector space
There is a quadratic form $q$ on a finite dimensional real vector space with associated symmetric bilinear form $\phi$. There is a subspace $R \leq V$ such that $\phi(r,v) = 0 \forall r \in R$ and $\...
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For a quadratic form q over the real numbers, is the set of vectors S such that, for all s in S, q(s) > 0, a subspace? If so, how do we prove it?
This question relates to the proof of theorem 11.1 of Chaper IV on page 168 of Evar Nering's "Linear Algebra and Matrix Theory", second edition. See e.g. here (page 182 of the pdf):
https://...
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Weight of a binary form
I'm going through P.J.Olver's book on classical invariant theory. He defines the action of $GL(2)$ on binary forms $Q(x, y)$ by a change of variables:
$$
\bar{x} = \alpha x + \beta y, \ \bar{y} = \...
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What is the surface area of the surface defined by $z = \frac{k_1}2 x^2 + \frac{k_2}2 y^2$?
What is the surface area of the 3D saddle surface $f$ defined by
$$
f = [x, y, z(x,y)]
$$
and with
$$
z(x,y) = \frac{k_1}{2} x^2 + \frac{k_2}{2} y^2
$$
for $(x,y)\in(-1,1)\times(-1,1)$ and for any $...
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Maximize sum of $(x_1+\dots+x_k)^2$ on the unit $n$-sphere
Given any positive integer $n$, let $t(n)$ be the smallest real number such that for any real numbers $x_1,\dots,x_n$, the following inequality holds $$ \sum\limits_{k=1}^n (x_1+\dots+x_k)^2 \leqslant ...
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Minimizing the Mahalanobis distance
Definitions
Consider the following optimization problem
\begin{equation*}\arg \min_{x\in\mathbb{R}^n} \lVert y-x\rVert_{P}^2\end{equation*}
where $y,P$ are given parameters and
\begin{equation*}
\...
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Bounds on the quadratic forms of a matrix using eigenvalues
I read from somewhere that the quadratic form of a matrix $A$, which is the $\vec{x}^\top A \vec{x}$ can be bounded by their eigenvalues:
$$\lambda_{\min} \| \vec{x} \|_2^2 \leq \vec{x}^\top A \vec{x} ...
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Proving that Every Quadratic Form With Only Cross Product Terms is Indefinite
I've looked into this statement for a while (in fact, maybe for too long), and I thought to myself that I had to prove it
If $Q \colon \mathbb{R}^3 \to \mathbb{R}$ is a quadratic form with only cross ...
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If a quadratic equation can have real and equal roots, then why don't we say it has one root?
Suppose $ax^2+bx+c$ is a quadratic equation with $D=0$
So it has the roots $x=\frac{-b}{2a},\frac{-b}{2a}$ which are real and equal
Why don't we just say it has one root which would be $x=\frac{-b}{...
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For a convex quadratic form $\frac{1}{2}h^{T}Ah+ b^{T}h+c$ , we have that the minimizer $y$ satisfies $y^{T}Ay = \max\{b^{T}h: h^{T}Ah \leq 1 \}^{2}$
I'm trying to follow the argument given on pg 24 of these notes. This is the proof of (2.11)
Here we are considering a convex quadratic form
$$f_{A,b}(h) = \frac{1}{2}h^{T}Ah+ b^{T}h+c$$
and I want to ...
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Bounding the inner product of a vector of correlations
Suppose $X$ is a Gaussian random variable and $Y$ is a Gaussian random vector of length $n$ (they are also jointly Gaussian).
Let $z$ be a vector with entries $z_i = \frac{\mathrm{Cov}[X, Y_i]}{\sqrt{\...
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Show the norm of projection decreases as the vector dimension increases
Given an $M\times L (M>L)$ matrix
$$\mathbf{A}=(\mathbf{a}_1,\mathbf{a}_2,\cdots,\mathbf{a}_L)$$
where
$$\mathbf{a}_k = (1,e^{j2\pi c\sin\theta_k},e^{j2\pi2c\sin\theta_k},\cdots,e^{j2\pi(M-1)c\sin\...
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Homogeneous quadratic parts of polynomials in Unbalanced Oil and Vinegar
In the book Multivariate Public Key Cryptography, the author describes the polynomials in the cryptographic system Unbalanced Oil and Vinegar in the following way:
Define $V=\{1,\dots,v\}$ and $O=\{v+...
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Criterion for unicity and existence of pre-image in multivariate cryptography
I am reading Ding's Multivariate Public Key Cryptosystems and in the book the author explains the so-called bipolar construction where one chooses three maps to construct encryption and signature ...
- 750
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All vectors that can be included in a basis such that a quadratic form is diagonalized in that basis
We have the quadratic form on $Q(x,y,z) = x^2 - y^2$ on $=\mathbb{R}^3$ considered as an $\mathbb{R}$-vector space. We want to calculate the set of vectors $v$ such that there exists a diagonal basis ...
- 430
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Finding all values for which the form is positive definite
Let $ q : \mathbb{R}^3 \to \mathbb{R}^3 $ be the quadratic form defined in the standard basis as $ q(x_1, x_2, x_3) = x_1^2+3x_2^2 +2x_1x_2+2x_2x_3 + 4\lambda x_1x_3 $ where $ \lambda \in \mathbb{R} $....
- 1,009
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votes
1
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Lower bounding the mean of a quadratic form with a positive semi-definite matrix
Let $\mathbf{X}\in\mathbb{R}^n$ be a vector random variable, and let $\mathbf{P}$ be an $n\times n$ positive semi-definite matrix.
I'm interested in deriving lower bounds on the expected value of the ...
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Why the factors of a quadratic equation $ax^2 + bx + c$ are given as $(x - \alpha)(x - \beta)$ and not as $(x + \alpha) (x + \beta)$
When reading about the quadratic equations, I came to the relation between the roots of the $p(x)$ with $a, b, c$.
Standard form of quadratic equation is: $ax^2 + bx + c$. Now, this is written in my ...
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1
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Can anyone show me $V(r)=V_0\ln(\frac{b}{a}) \ln(r)-V_0\ln(\frac{b}{a}) \ln(b)=\frac{V_0}{\ln(\frac{a}{b})}\ln(\frac{r}{b})$?
There is an equation, $V(r)=C_1 \ln(r)+C_2$, now I know $V(b)=0$, and $V(a)=V_0$, find $C_1$ and $C_2$.
$V(b)=C_1 \ln(b)+C_2=0$
$V(a)=C_1 \ln(a)+C_2=V_0$
$V_0=C_1(\ln(a)-\ln(b))=C_1\ln{\frac{a}{b}}$, ...
- 15
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votes
1
answer
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Inequality about matrix quadratic form
Consider a vector $x \in \Bbb R^n$ and two positive definite matrices $A, B \in \Bbb R^{n\times n}$ satisfying a linear matrix inequality $A - B > 0$. Can we draw the conclusion that $x^\top Ax >...
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0
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Find a tighter upper bound for the ratio involving quadratic forms
For an $M\times M$ matrix
$$\mathbf{R}=\mathbf{I}+\sum_{k=1}^{L}\lambda_k^2\mathbf{a}_k\mathbf{a}_k^H=\mathbf{I}+\mathbf{A}\mathbf{\Lambda}\mathbf{A}^H$$
where $\mathbf{I}$ is an identity matrix, $\...
- 399
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votes
0
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Branching out domain topographs
I'm reading an illustrated theory of numbers by Martin H. Weissman, on page-233, it is shown how to expand out a domain topograph starting from basis $\pm (17,12)$ and $\pm(7,5)$
I understand that ...
- 11.6k
4
votes
1
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Prove that $\sum_{i,j=1}^n \frac{a_ia_j}{1 - a_i^2a_j^2}\geq 0$ where each $|a_i| < 1$
This question is inspired by Prove that $\prod_{1\leq i,j\leq n}\frac{1+a_ia_j}{1-a_ia_j}\geq1$ for $n$ real numbers $a_i\in(-1,1)$.
Let $(a_i)$ be a sequence of real numbers that satisfy $0 < |a_i|...
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Does $\alpha \in G_F(\pi)$ where $\pi \cong \pi' \perp b \pi'$ imply $\alpha (\pi' \perp \langle b \rangle) \cong \pi' \perp \langle b \rangle$?
Let $\pi, \pi'$ be some Pfisterforms with $\pi \cong \pi' \perp b \pi'$ for some $b \in F^*$ and $\alpha \in D_F(\pi) = G_F(\pi) = \{ x \in F^* \mid x \pi \cong \pi \}$. So we have $\alpha \pi \cong \...
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Independence of two quadratic forms when the population is multi-normal.
Suppose $X \sim N_n(\mu, \sigma^2 I_n)$ and let $A$ and $B$ symmetric with shape $n\times n$, prove that $X^TAX$ is independent of $X^TBX$ if $AB=0$.
My attempt is that I have shown that $X^TAX$ is ...
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