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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
31 views

How to Compute the Second Derivative of a Quadratic Maximization Problem with Respect to the Weight Parameter?

I am working on a problem involving the maximization of a function subject to a normalization constraint, specifically in the context of quadratic forms and eigenvalues. The objective function is ...
Krishnendu K's user avatar
-1 votes
0 answers
14 views

Lower bound for the formula

Is there any formula to find the lower bound or the approximation of the below? $$ \left(\frac{y+\frac{k}{y}}{x+\frac{k}{x}} \right)^2 - \left(\frac{y}{x}\right)^2 $$
hlm's user avatar
  • 1
0 votes
1 answer
25 views

Quadratic form with positive semi-definite matrix

Suppose $A$ is a positive semi-definite matrix, and $x$ and $y$ are real vectors. Under what conditions does the following result hold? $$ x'y > (<) 0 \Longrightarrow x'Ay \geq (\leq) 0 $$ ...
HXW's user avatar
  • 85
4 votes
1 answer
163 views

Ratio of cubic and quadratic form is approximately normal?

Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which is a ratio of a cubic form and a ...
rgvalenciaalbornoz's user avatar
0 votes
0 answers
27 views

Sesquilinear generalization of symplectic form

I have recently been trying to learn some basic symplectic geometry, and I have come across two sesquilinear forms which are closely related to the symplectic form. Fix $\mathbb K$ to be a field, and $...
Cole Comfort's user avatar
0 votes
1 answer
75 views

Positiveness of Bilinear form

I would like to show that $$A = x^T G(x) \tanh(x) \geq 0$$ (with equality only for $x=0$), for $x\in\mathbb{R}^n$. It is known that $G(x)\in\mathbb{R}^{n\times n}$ is a symmetric and positive definite ...
Chris's user avatar
  • 33
0 votes
0 answers
28 views

Seeming contradiction between Genus theory on quadratic fields, and computations of sizes of class groups for those quadratic fields

The Fundamental Theorem on Genera of Quadratic Fields (Satz 100 in Hilbert's Zahlbericht) says as follows: "an arbitrary set of $r$ units $\pm 1$ is the character set of a genus of the field $\...
pedroelpanda's user avatar
0 votes
0 answers
21 views

Eigenvalues of a symmetric matrix with known column/row $l^2$-norm

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Suppose we use the notation $a_{i}$ for the $i$-th row (or column, which would be the same given its symmetry), with $i\in\{1, ... , n\}$. ...
controllystuff's user avatar
4 votes
5 answers
108 views

Let $n\equiv 1\pmod 8$. Do there exist $x,y,z\in\mathbb{Z}$ with $x\equiv \pm3\pmod 8$ and $x^2+4y^2+4z^2=n$?

Let me preface this by saying I know very little about quadratic forms and most of what I know is about quadratic forms in two variables, whilst this question is about a quadratic form in three ...
Mastrem's user avatar
  • 8,266
1 vote
0 answers
67 views

Prove this is a bilinear form

I'm dealing with an exercise on bilinear forms, but I don't know how to properly start. I have to prove that $\beta(A, B) = \mathrm{tr}(AB) - \mathrm{tr}(A)\mathrm{tr}(B)$ is a bilinear form, where $A,...
Martin and Friends's user avatar
0 votes
1 answer
57 views

Why are these two quadratic forms the same even though I'm changing the associated random vector?

To give you some context first: Let the sample variance $S^2$ be $S^2 = \frac{\sum_i (\textbf{Y}_i - \overline{\textbf{Y}})^2}{n-1}$, with $\textbf{Y}\sim N_n(0,\sigma^2\textbf{I}_n)$. From a previous ...
l337n00b's user avatar
  • 475
1 vote
0 answers
75 views

Multiply two lines together

Excuse me , I had a strange question :). In algebraic geometry, a quadratic expression can be written as the product of two different linear equations, for example: y = x² + 3x - 4 can be factored as ...
Mostafa Zeinodini's user avatar
1 vote
2 answers
102 views

Derivative of quadratic matrix product (not vector $d (xAx^T) = x (A + A^T))$, $A$ is not symmetric

I am trying to differentiate the matrix product of the quadratic form, $M=XUX^T$, with respect to the matrix, $X$. That is, I want $\frac{\partial M}{\partial X}$. Working from a concrete example ...
bacchus's user avatar
  • 11
1 vote
0 answers
42 views

Is it Possible to Treat a Quadratic System of Equations as Linear to Show that the Equations are not Independent?

I have a system of $N$ equations with $n$ complex variables. Each equation has the form $a x_n^2 + b Re(x_nx_{n-1}) + ...+ c x_{n-1}^2 + d Re(x_{n-1}x_{n-2}) +...+ fx_1^2 = H$, where $H$ is a real ...
skeer16's user avatar
  • 21
2 votes
1 answer
89 views

If $\alpha$ is a root of the equation $x^{2}+x+1=0$ and $(1+\alpha)^{7}=A+B\alpha+C(\alpha)^{2},$ then find the value of $5(3A-2B-C)$

Q)If $\alpha$ is a root of the equation $x^{2}+x+1=0$ and $(1+\alpha)^{7}=A+B\alpha+C(\alpha)^{2},$ then find the value of $5(3A-2B-C)$ Ans) My Approach: I considered $\alpha=\omega$. Because we know ...
Deb Sankar Roy's user avatar
1 vote
0 answers
25 views

Name for $N^T A N$ when N is rectangular?

According to wikipedia, a transformation of the form $$A → P^T A P$$ is called a congruence transformation when $P$ is invertible (and thus square). I often run into transformations of the similar ...
Alec Jacobson's user avatar
0 votes
0 answers
38 views

If a quadratic form $Q:\ V\longrightarrow\mathbb R$ is non-vanishing except at $0$ then $Q$ is positive or negative definite

Let $Q:\ V\longrightarrow\mathbb R$ be a quadratic form such that $Q(u)\ne 0$ for all $u\ne0$. Then $Q$ is positive or negative definite. From the definition, I need to show $Q(u)>0$ (or $<0)$ ...
PermQi's user avatar
  • 409
0 votes
0 answers
20 views

if a quadratic form $Q$ is positive definite then $\big|[Q]-\lambda I_n \big|$ has non-zero alternating sign coefficients

Let $Q:\ \mathbb R^n\longrightarrow\mathbb R$ is a positive definite quadratic form and $\mathcal B$ is a basis of $\mathbb R^n$. Then how to show that all coefficients of the characteristic ...
PermQi's user avatar
  • 409
0 votes
0 answers
24 views

If the matrix of the quadratic form $Q$ in some basis is $A^\top A$ then $Q$ is positive definite

Let $A\in GL_n(\mathbb R)$ and $Q:\ \mathbb R^n\longrightarrow\mathbb R$ is a quadratic form such that $\big[Q\big]_{\mathcal B}=A^\top A$ for some basis $\mathcal B$ of $\mathbb R^n$. Then $Q$ is ...
PermQi's user avatar
  • 409
1 vote
2 answers
58 views

If a sequence of quadratic forms converges pointwise, then does the associated sequence of matrices also converge?

Consider the sequence of functions $f_k : \mathbb N \times \mathbb R^n \to [0,\infty)$ defined as $f_k(x) = x^TP_kx$, where $P_k : \mathbb N \to \mathbb S_+^n$ is a sequence of symmetric and positive ...
mhdadk's user avatar
  • 1,391
5 votes
3 answers
132 views

Find the lengths of the principal axes of the ellipsoid $\sum_{i \leq j} x_ix_j = 1$ (Arnold 85)

Find the lengths of the principal axes of the ellipsoid $$\sum_{i \leq j} x_ix_j = 1.$$ -- Arnold, Trivium 85 My solution is below. I request verification, feedback, or alternate approaches (...
SRobertJames's user avatar
  • 3,958
4 votes
1 answer
195 views

On the elliptic curve $X^3 + 6\cdot 163^2 X - 7\cdot 163^3 = Y^2$ and others

I. Ellipse Given the general equation, $$a^3 + b^3 + c^3 = (c + m)^3$$ Let, \begin{align} c &= (n + 1)(a - m) + n b\\ a &= p + q + m + 4 m n + 3 m n^2\\ b &= p - q + 2 m n + 3 m n^2 \end{...
Tito Piezas III's user avatar
0 votes
0 answers
46 views

Quadratic forms in three variables

In my self-study of quadratic forms I am using J.W.S. Cassels' Rational quadratic forms (1978) as my main source of wisdom. For some time now I have been stuck in the chapter that covers ternary forms,...
Marconi_1900's user avatar
1 vote
1 answer
88 views

A question related to the definition of quadratic form.

Let $V$ be a finite dimensional vector space over the field $\Bbb F$ ($\Bbb R$ or $\Bbb C$) and let $q: V\to\Bbb F$ be a mapping. Then is it true that the following two conditions are independent, ...
Mrityunjay Pandey's user avatar
1 vote
0 answers
61 views

What is the name of the middle matrix in a quadratic form $x'Ax$?

In a quadratic form, $x'Ax$, does matrix $A$ have a special name? I am trying to write a sentence that looks like "Using vector $x$ and matrix $A$ as the ______ matrix, we can write the quadratic ...
Adi's user avatar
  • 19
1 vote
1 answer
83 views

Single constraint quadratic optimization dual form expression using the Schur complement

Strong duality result for non-convex problem with two quadratic functions is a related question. However, I am trying to understand how the dual form problem comes about. This dual form ...
FXQuantTrader's user avatar
0 votes
0 answers
33 views

Help in understanding why the antisymmetric part of the positive semi-definite quadratic form does not contribute?

Can you help me understand the statement given below, particularly what is said when evaluating (15) and the conclusion leading to, and given by, (16)? On a mathematical level, in view of the scalar ...
Armadillo's user avatar
  • 525
3 votes
2 answers
144 views

Are there any efficient methods to manually factor a quadratic in two variables?

For a quadratic in two variables, $ax^2+bxy+cy^2+dx+ey+f$ may be factored into the form $(p_1x+q_1y+r_1)(p_2x+q_2y+r_2)$ given that $\det\begin{bmatrix}a & b/2 & d/2\\ b/2 & c & e/2\\d/...
Catherine's user avatar
0 votes
0 answers
38 views

Minimizing quadratic cost function over an Euclidean ball

Let $f : \mathbb{R}^d \to \mathbb{R}$ be defined by $$f(\theta) := \| \theta - z \|^2_\Delta := (\theta-z)^t \Delta (\theta-z)$$ where $\Delta = \operatorname{diag}(s_1, \ldots, s_d)$, where $s_i > ...
deque's user avatar
  • 551
0 votes
0 answers
50 views

In the perspective of tensors, are the matrices in linear transformations and quadratic forms fundamentally different?

I do not understand the full theory of tensors, but have heard several things about them: Linear transformations are (1,1)-tensors. Metrics are (0,2)-tensors. Quadratic forms and metrics look very ...
ZhenRanZR's user avatar
1 vote
2 answers
169 views

Approaches to Solving Quadratic Diophantine Equations of the Form $x^2 + y^2 = k$

I am exploring the solutions of quadratic Diophantine equations in the following form: $$ x_1^2 + y_1^2 = x_2^2 + y_2^2 = \dots = x_n^2 + y_n^2 = k, $$ where $x_1, \dots, x_n, y_1, \dots, y_n, k \in \...
Ba_nanza's user avatar
0 votes
0 answers
58 views

How can I tell if these matrices are congruent?

I am completey lost on this. I know a matrix $B$ is congruent to $A$ if $B = P^\top\!\!AP$ but I tried finding the e-vectors and e-values for $P$ and $P^\top\!$ to spit out $A$ again. I really don't ...
Dream Cloud's user avatar
1 vote
0 answers
32 views

Why is the centered moment for this formula so different?

Define the following two functions $Q$ and $S^2$: One of the exercises now asks me to calculate the variance of $S^2$. I tried doing so with the help of a formula that makes use of the third and ...
l337n00b's user avatar
  • 475
0 votes
0 answers
21 views

Existence of solutions to system of quadratic equations

I am currently trying to understand when the following system of equations has a solution: $$x^t U_i^\dagger U_j x = c_{ij}$$ Where the $U_i$ are $n$ by $n$ unitary matrices, $x \in \mathbb{C}^n$, $c_{...
burofiz's user avatar
0 votes
1 answer
33 views

signature of quadratic form and block matrix

Suppose, there is a quadratic form with matrix $A = A^T$ and signature $(p,q)$. How can I find the signature of quadratic form with matrix $$ \begin{pmatrix} A & A \\ A & A \\ \end{pmatrix} $$ ...
GIFT's user avatar
  • 321
0 votes
1 answer
61 views

$2×2 $ real symmetric matrix full classification [closed]

Let $$A=\begin{pmatrix}a&b\\b&c\end{pmatrix}$$ all real numbers. I know that If $\operatorname{det}(A) > 0$ then $A$ is positive definite if $a > 0$ and negative definite if $a < 0$. ...
root's user avatar
  • 153
3 votes
1 answer
45 views

Understanding what 'LLL-reduced in the direction of v' means

I've been working with ideal reduction algorithms, and in particular have a need to understand the notion of LLL-reduced along a vector $v$, and in particular what it actually means to have 'small $v$-...
Punchinello's user avatar
1 vote
1 answer
51 views

Derivation of a formula for the quadratic form of a matrix

Reading through Seber's Linear Regression Analysis, I came across the following formula for the quadratic form of a matrix: Suppose $\textbf X$ was a $n \times 1$ vector of random variables, suppose $\...
l337n00b's user avatar
  • 475
1 vote
2 answers
86 views

Linear Regression Seber - Why is the matrix A written in this way?

Example 1.9 of Seber’s book states the following: The result stating that if $\rho = 0$ it holds that $Q = \sum _i (X_i - \overline X)^2$ has expected value $\sigma ^2 (n-1)$ follows from $\...
l337n00b's user avatar
  • 475
1 vote
1 answer
66 views

Solving $X A X^T = I$ subject to $b X = B$

I am facing the following system of matrix equations: $$ b X = B $$ $$ X A X^T = I $$ where $X$ is the square matrix of dimension $N \times N$ to solve for, $b$ and $B$ are both row vectors of ...
Quantuple's user avatar
  • 148
0 votes
0 answers
18 views

Isotropic vectors of Tits form

Let $\Gamma=(\Gamma_0,\Gamma_1)$ be a multigraph, that is, $\Gamma_0$ is a (finite) set of vertices and $\Gamma_1$ is a (finite) multiset of edges. This means we allow multiple edges between two ...
Albert's user avatar
  • 2,722
1 vote
1 answer
40 views

Find a matrix such that the quadratic form of a orthonormal basis is equal to the Kronecker delta

For $n \in \mathbb{N}$ let $\{v_1, v_2, \ldots v_n \}$ be a orthonormal basis of $\mathbb{R}^{n}$. Further, let $i \in \{ 1, 2, \ldots, n \}$ be arbitrary but fixed. I am trying to prove that there ...
SebastianP's user avatar
4 votes
1 answer
228 views

Algebraic relations between alternative representations of an integer in the form $p^2+3q^2$

In the course of pondering this question I noticed that $4[(c+1)^3 – c^3]$ can be represented in the form $p^2 + 3q^2$ in three different ways: $$4[(c+1)^3 – c^3] = 12c^2 + 12c + 4$$ $$= 1^2 + 3(2c+1)^...
Adam Bailey's user avatar
  • 4,132
0 votes
0 answers
28 views

Relate solutions to quadratic eqs.

I have two quadratic eqs: \begin{align} \frac1{x} & = 2 + \frac1{1-r} + \frac1{1-x}\\ \frac1{y} & = 2 - \frac1{r} + \frac1{1-y}, \end{align} where $r\in(0,1)$ is a parameter. I am interested ...
Morad's user avatar
  • 605
0 votes
1 answer
30 views

Rank-24 Leech matrix cannot have simultaneous integer entries with unit determinant or integer determinant?

A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix: ...
zeta's user avatar
  • 191
-1 votes
1 answer
80 views

When does a quadratic inequality have a solution? [closed]

Consider the following quadratic inequality: $$x^T Q x + c^T x \leq b$$ for $x\in \Re^{d}$. When does this system have a solution? Without loss of generality assume that $Q$ is symmetric. Looking for ...
AspiringMat's user avatar
  • 2,288
0 votes
0 answers
64 views

Weird quadratic form exercise

Let $H$ be the quadratic form in vector space $\mathbb{R}^3$ s.t $$H=x^2+2y^2+3z^2+2xy+2xz+2yz.$$ Let $\varphi$ be the symmetric bilinear form on $\mathbb{R}^3$ and $\varphi:\mathbb{R}^3\times \mathbb{...
Harry's user avatar
  • 161
1 vote
0 answers
42 views

Quadratic Form and its Matrix, Associated bilinear form for positive definite quadratic form is nondegenerate.

I have a question regarding the quadratic forms and their associated matrices. For some reason, google keeps telling me that this matrix is symmetric. The definition I am using is that a quadratic ...
Goob's user avatar
  • 391
2 votes
1 answer
132 views

Some basics on octonions and quaternions

There is an $\mathbb{R}$-bilinear operation on the octonions $\mathbb{O} \otimes_{\mathbb{R}} \mathbb{O} \rightarrow \mathbb{O}$, which is not associative. My question is instead about the algebra ...
Ronald J. Zallman's user avatar
1 vote
0 answers
22 views

Parameterization of transformation preserving norm in R^n

I am interested in the diffeomorphisms $T : \mathbb{R}^n \to \mathbb{R}^n$ that preserve the Euclidian norm, i.e such that $T(x)^\top T(x) = x^\top x$ for all $x \in \mathbb{R}^n$. Do we know how to ...
Mathieu le provost's user avatar

1
2 3 4 5
48