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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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Write $\frac{ X^{2} + Y^{2} }{ XY }$ in Quadratic Form

Is there a way to write $$ \frac{X^{2} + Y^{2}}{XY} $$ in quadratic form. I am struggling to set up the proper vector and find a square matrix.
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on the quadratic form $5x^2-y^2$

Consider the following subsets of $\mathbb{Z}$: $A=\{\frac{5p^2-y^2}{4}\, | \, p, y \,\,\text{odd positive integers}, p\,\, \text{prime}\}$ and $B=\{\frac{5x^2-y^2}{4}\, | \, x, y \,\,\text{odd ...
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Nonlinear Recursion Solution Process for $x_{n+1}=\Sigma_{i=1}^{n} x_{i}x_{n-i}$ (Known Solution)

I want to solve the equation $x_{n+1}=\Sigma_{i=1}^{n} x_{i}x_{n-i}$. Plugging the equation into Mathematica gives me $x_n=(-1)^{n}2^{2n+1} Binomial(1/2, n+1)x_0^{n+1}$. How might I derive this?
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Creating a quadratic equation from a condition

how can I create an equation that satisfies the following: "x-intercepts 1 and -1, y-intercept 3". I understand that the factored form of a quadratic equation offers both x intercepts however, I'm not ...
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The square trinomial $y=ax^2+ bx + c$ has no roots and $a + b + c > 0$. Find the sign of the coefficient $c$ .

The square trinomial $y=ax^2 + bx + c$ has no roots and $a + b + c > 0$. Find the sign of the coefficient $c$. I'm having difficulties with this problem. What I've tried: I realized that a ...
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Orbits of the conjugation action of $GL_3 (\mathbb{R})$ on the nonsingular symmetric $3\times3$-matrices

Let $S$ be the space of all symmetric $3 \times 3$ matrices of full rank and with real entries. $GL_3 (\mathbb{R})$ acts on this space by conjugation, \begin{align*} g.A = (g^{-1})^T A g^{-1}, \quad ...
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Linear Algebra - Positive-Definiteness in Vectors?

I was reading up on the inner product over at this Wikipedia page, and I noticed, in the given definition, the use of the term "positive-definiteness". Now, from what I know, this is terminology one ...
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2answers
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Converting standard form to vertex form, parental homework help

I'm trying to help my son with his homework but am having trouble feeling confident that I know what the assignment is asking for. I've been learning (maybe relearning) about standard vs vertex form ...
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1answer
36 views

Sign of quadratic form with parameter

Given $Q(x,y,z;\alpha)=x^2+z^2+2\alpha xy+2xz$, i have to study the sign of quadratic form. Obviously i can use eigenvalues or studying the sign of minors, but in this case i have a hard time to ...
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Equivalence of quadratic forms over arbitrary field (char(K)≠2)

Let $\mathbb{K}$ be an arbitrary field with $char(\mathbb{K})\neq 2$ and let $a_1, \ldots ,a_n$ be Elements of $\mathbb{K}$. I want to proof the following equivalence of $n-$quadratic forms in ...
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Inequality with roots and bivariate quadratic forms

For $|a|\le 1$ and $|b|\le 1$ show that $$\sqrt{1 - b^2}\sqrt{1 - a} \le \frac{\sqrt{6}(1-\frac{3}{4}b)(7a^2b^2 + 96a^2b + 65a^2 + 14ab^2 + 42a + 7b^2 - 96b - 135)}{43a^2b^2 + 129a^2 + 86ab^2 - 86a ...
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How can I guarantee the existence of a solution to this quadratic system of equations?

I have $n$ real quadratic equations and $n$ real variables, $x_i$, of the following form: $$\sum_{i\neq j} a_{ijk}x_ix_j+\sum_ib_{ik}x_i+c_k=0 \ \forall k$$ for $i,j,k\in\{1,\dots n\}$; all ...
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Is $X + \frac{2}{X}$ a rational quadratic form, where $X \in \mathbb{N}$?

First of all, I apologize for the rather silly question. This came up while I was scouring the Internet on a mathematical terminology appropriate for a concept that I need for a paper which I am ...
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If $p$ is prime, then $x^2 +5y^2 = p \iff p\equiv 1,9 $ mod $(20)$.

Let $p\neq 2,5$ be prime. I wish to show that: $x^2 +5y^2 = p \Leftrightarrow p\equiv 1,9 $ mod $(20)$. I proved to $\Rightarrow$ part, means $x^2 +5y^2=p \Rightarrow p\equiv 1,9 $ mod $(20)$. For $\...
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Pseudo convex quadratic forms are quasi convex

I have been trying to show: A Pseudoconvex quadratic form is Quasiconvex in the same set X, by using the following definitions. These definitions are equivalent to the standard definitions of Pseudo ...
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1answer
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$x^2 + 3xy + y^2 = n$ Diophantine Equation

I was wondering if someone could direct me towards information regarding the $x^2 + 3xy +y^2 = n$ diophantine equation. Additionally, is there anything about the general case of these diophantine ...
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Explicit matrix representing composition of binary quadratic forms

A 2x4 matrix $M$ can be viewed as defining a linear operation mapping two 2x2 matrices to one: $A \times B \rightarrow C$. Breaking $M$ into 2x2 blocks this can be written as $$f(A,B) = \begin{bmatrix}...
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make c the subject $c+1=\sqrt{a-ac^2}$

Transpose the formula to c the subject, $$c+1=\sqrt{a-ac^2}\tag1$$ A possible method $$(c+1)^2=a-ac^2$$ $$c^2+2c+1+ac^2=a$$ $$c^2(1+a)+2c+1=a$$ I can't seem to make c the subject. I could use ...
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Simultaneous diagonalisation of quadratic forms

Is there a linear transformation that simultaneously reduces the pair of real quadratic forms $$x^2-y^2$$ and $$2xy$$ to diagonal forms? My attempt I know that neither of these forms are positive ...
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Find a diagonal form of the quadratic form $f(x) = \sum_{i = 1}^nx_i^2 + \sum_{i < j}x_ix_j$

Find a diagonal form of the quadratic form $$f(x) = \sum_{i = 1}^nx_i^2 + \sum_{i < j}x_ix_j.$$ It turned out to be such a problem: How to change quadratic form $f(x) = \sum_{i = 1}^nx_i^2 + \...
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How did Euler show that number is idoneal?

Euler famously showed that there are at least 65 idoneal (convenient) numbers. This was Euler's definition of idoneal number: Number $n$ is idoneal if following holds: Let $m>1$ be an odd number ...
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1answer
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Value of the quadratic form as a function of the determinant

Suppose I have a symmetric, positive definite matrix $A$ with all the diagonal elements $\sigma^2 > 0$, and its off-diagonal elements are $\rho \in [-1,1]\backslash\{0\}$. Consider a quadratic ...
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properties of Quadratic Gauss sum

Proof that $$ G(n,p^k)=pG(n,p^{k-2}) \ \ if \ \ k ≥ 2 \ \ and \ \ p \ \ is \ \ an \ \ odd \ \ prime \ \ number \ \ or \ \ if \ \ k ≥ 4 \ \ and \ \ p = 2. $$ Where $$ G(n,m)=\sum_{x=0}^{m-1}e\left(\...
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35 views

Quadratic form as a homogenous polynomial

Let $Q(v)=v'Av$, where$v=\begin{pmatrix}x&y&z&w\end{pmatrix}\ \ A=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}$, then does ...
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Understanding Exercise 3.12 in Cox “Primes of the form $x^2+ny^2$”

I'm afraid this will only make sense if you have access to the book, as it's far too complicated to retype all the prerequisites. I'm trying to understand what Exercise 3.12 in Cox "Primes of the ...
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Why is the polarization identity important, intuitively?

The polarization identity states, roughly, that a norm satisfying the parallelogram law induces a vector space inner product (and vice versa). This has many nice applications, such as a simple ...
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A binary quadratic form whose range topograph has a lake

Is it true that if Q is a binary quadratic form whose range topograph has a lake, then Q factors as a product of two linear forms with integer coefficients?
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Do two equivalent quadratic forms necessarily have the same solutions

Do two equivalent quadratic forms necessarily have the same solutions? Suppose that I have $Q(x,y)= x^{2}- xy+ 8y^{2}$ and $R(x,y)= 2x^{2}+ 3xy+ 5y^{2}$ and the value of $Q(2,1)$ and $R(2,1$) are ...
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Intervals of a Multivariable Function

If the gradient at some point of a multivariable function equals $\vec{0}$, and the Hessian is positive or negative semidefinite, is there a notion, as in single variable calculus, of resolving the ...
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Necessary and sufficient conditions for $x'Ax = 0$

I came across the following problem and I am having a hard time thinking about it. Let $A$ be a $k\times k$ real matrix. Notice that I do not require that $A$ is symmetric, positive definite or ...
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Finding the minimum value.

I'm struck on this question, I tried hard but couldn't solve it. Question: if a quadratic equation in $x$: $$ax^2 - bx + 5 = 0$$ does not have two distinct real roots, then find the minimum value of $...
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$\{x\mid x^TAx\leq 1\} = \{x\mid x^TBx\leq 1\} \Rightarrow A = B$, where $A\succ 0, B\succ 0$

I want show that $$\{x\mid x^TAx\leq 1\}_{\epsilon_A} = \{x\mid x^TBx\leq 1\}_{\epsilon_B} \Rightarrow A = B,$$ where $A\succ 0, B\succ 0$ (positive definite). Can I prove this by arguing the ...
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1answer
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Factoring Determinant

I am trying to find the determinant of this matrix with eigenvalues in it. $(\lambda I - A)$ = $\begin{bmatrix} \lambda - 1 & 3 & 0 \\ 3 & \lambda - 1 & 0 \\ 0 & 0 & \lambda + ...
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Deriving the Transformation of the Coefficients of Surfaces of Second Degree as a Tensor. Per Einstein

See pages 12 and 13 of Einstein's https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1 This question involves rectangular Cartesian coordinates in $\mathbb{R}^{3}$ related by orthogonal ...
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When do integers satisfy simple quadratic relation?

$$((w+x)(y+z)-(w-x)(y-z))^2=4(wy+xz)^2-4(w^2-x^2)(y^2-z^2)$$ is true and so if $(a-b)^2=4(e+f)^2-4ab$ then under what additional minimal conditions can we say $e=wy$ and $f=xz$ while $a=(w+x)(y+z)$ ...
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1answer
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More questions on quadratic forms over field

I'm a student, currently studying about quadratic forms over a field $\mathbb{R}$ and I have a few questions regarding the topic. From a book I currently read, a quadratic form is a real-valued ...
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Quadratic Forms rank and signature confusion

I would just like to clear some things up! A quadratic form can always be expressed in terms of a symmetric matrix. When we diagonalise this matrix we can read the rank and signature of the quadratic ...
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Solving a system of ternary quadratic equation

I'm wandering if there are any appropriate ways to solve this system of ternary quadratic equations for $x,y,x$. \begin{equation} \left\{ \begin{aligned} x^2+xy+y^2 & = s(x+y) \\ y^2+yz+z^2 & ...
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Sum of degenerate quadratic forms.

I am searching for an analogue of the fact: let $\Sigma_1 , \Sigma_2> 0$ in $\mathbb R^{m \times m}$ and let $x,c_1, c_2 \in \mathbb R^m$ be arbitrary. Let $\Sigma_3^{-1} = \Sigma_1^{-1} + \Sigma_2^...
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A problem on positive definite matrix

I am studying a chapter on positive definites and there is this question in which I have to find whether the quadratic forms are positive definite or not. I have to confirm my answer for this ...
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Substitution to get rid of cubic terms in a two-variable expression

Consider the quadratic form $$Q(u,v) = au^4 + bu^3v+cuv^3+du^2v^2+ev^4$$ If $a,b,c,d = 1$ then we can get rid of the cubic terms by substituting $u:=x-y, v:=x+y$. But what would be the general ...
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How to find image of a quadratic form?

Suppose the quadratic form $f: \mathbb R^3\to \mathbb R$ with $$f(x_1,x_2,x_3) = x_1^2 - x_2^2 - 11x_3^2 - 2x_1x_2 + 4x_1x_3 + 8x_2x_3.$$ By using Lagrange's Reduction, we have the canonical ...
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Necessary and sufficient condition for quadratic forms

Consider the quadratic form $Q(u,v)=au^2 + 2buv +cv^2$. Upon completing the square we will obtain $$Q(u,v)=a\left(u+\frac{bv}{a}\right)^2+\left(c-\frac{b^2}{a}\right)v^2,$$ we assume that $a\ne 0$. ...
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Completing the square to decompose quadratic forms in three variables

Given the quadratic form $Q(\textbf{x}) = x^2 + 2xy - 4xz +2yz -4z^2$, the question asks to decompose $Q$ into sums of squares, first by eliminating terms in $z$, then terms in $y$, and then terms in $...
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Reduced equation of quadric surfaces

Given the following quadric surfaces: Classify the quadric surface. Find its reduced equation. Find the equation of the axes on which it takes its reduced form. The quadric surfaces are: (1) $3x^2 +...
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MIQP problem slow to solve: how to rewrite it?

I am looking for suggestions on how to rewrite a MIQP problem. Let me firstly introduce the problem Notation: The unknown vector is $x$ with size $(4*2+225*2)\times 1$. We can think of the ...
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30 views

Signature of matrix associated with q

For $\alpha\in\mathbb{R}$, let $q(x_1, x_2) = x_1^2 + 2\alpha x_1x_2 + \dfrac{1}{2}2x_2^2$, for $(x_1, x_2) \in \mathbb{R^2}$.Find all values of $\alpha$ for which the signature of $q$ is 1. The ...
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24 views

Min and max eigenvalues of a quadratic form

I have the following quadratic form : $q(x,y,z)=x(y+4z)+y(x-2z)+x^2$, represented by the symmetric matrix $\begin{bmatrix}1&1&2\\1&0&-1\\2&-1&0\end{bmatrix}$. I am asked to ...
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57 views

Signature of a quadratic form from matrix

I have the following quadratic form : $q(x,y,z)=x(y+4z)+y(x-2z)+x^2$. Before giving its signature, I am asked to work out its matrix, which should be $\begin{bmatrix}1&1&2\\1&0&-1\\2&...
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1answer
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Is there only one unique minimizer for this scalar complex quadratic form?

Consider $c,b \in \mathbb{C}$ and $f: \mathbb{C} \mapsto \mathbb{R}$, $$f(c) = bc' + b'c + cc'$$ is there only one extrema of $f$ corresponds to $$c^* = -b$$? where, $'$ means complex conjugate. ...