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.

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12 views

Is every quadratic form in characteristic $2$ induced by a bilinear form?

Let $F$ be a field of characteristic $2$, and $q$ a quadratic form on $F^n$ for some positive integer $n$, i.e., $q:F^n\rightarrow F$ is a function such that $g(x,y):=q(x+y)-q(x)-q(y)$ is a bilinear ...
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If $x^{T}A^{T}Ax = x^Tx$ holds for every $x$, then $A^{T} A = I_n$

Given $A \in \mathbb R^{n \times n}$, if $$\left( \forall x \in\mathbb R^n \right) \left(x^{T} A^{T} A x = x^T x \right)$$ how to conclude that $A^{T}A = I_n$? I appreciate any help!
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Can this be an onto function?

A function $f:R\to R$, where $R$ is set of real numbers, is defined by $$f(x)=\frac{\alpha x^2 + 6x - 8}{\alpha + 6x -8x^2}$$ Find the interval of values of $\alpha$ for which $f$ is onto. Justify ...
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Proof check: Using Hanson-Wright inequality to concentrate a quadratic form $y^\top A y$ where both $y$ and $A$ are random but independent

Disclaimer. I don't know if this is the right venue to ask this. I'm working out a bigger proof, in a critical step, I'ved used an argument I'm not quite sure about. Let $n$ be a large positive ...
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when quadratic form restricted to tangent plane of surface is degenerate

Consider the one-sheeted hyperbola $X= \{ x^2 -y^2 -z^2=-1 \}$ with quadratic form $q(x,y,z) = x^2 -y^2 -z^2$. I want to understand when $q$ restricted to the tangent plane $T_pX$ for $p \in X$ is ...
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59 views

Partial derivatives of $|Ax - b|^2$? [duplicate]

I'm trying to work out the partial derivatives of a function $L$ in terms of $x_i$: $$ A \in \mathbb{R}^{m x n} \quad b \in \mathbb{R}^m \quad x \in \mathbb{R}^n $$ $$\begin{aligned} L(x) &= \left\...
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Quadratic forms for unbounded operator

Let $A$ be an unbounded, self-adjoint operator on Hilbert space $H$. Let us pass to a spectral representation of $A$, so that $A$ is multiplication on $x$ by $\bigoplus_{n=1}^NL^2(\mathbb{R},\mu_n)$. ...
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Eigenvalues of solution to matrix Riccati equation

I am interested in an upper and lower bound on the eigenvalues of $X$, where $X$ satisfies $$ X^{-1} A X^{-1} = B,$$ for symmetric and positive semi-definite matrices $A \in \mathbb{R}^{n \times n}$ ...
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Rational solutions of quadratic forms

Is there an algorithm or a method that one can use to determine whether an equation of the form $(\text{E})$: $$ax^2+by^2+cz^2+dt^2=0$$ has a solution $(x,y,z,t)$ in whole numbers. In other words, ...
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complete the squares to find expression of quadratic forms

I don't understand what is the intuition/ strategy to obtain $q(u) = \frac{1}{4}(u_1 + u_2 + 2u_3)^2 - \frac{1}{4}(u_1-u_2)^2-u_3^2$ from $q(u) = u_1u_2 + u_1u_3 + u_2u_3.$ I understand the result of ...
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With $n> 3,$ prove that $a_{1}+ a_{2}+ a_{3}\geq 100$ by using Karamata's inequality

Given $n$ real numbers $a_{1}, a_{2}\cdots a_{n}$ so that $$a_{1}\geq a_{2}\geq\cdots\geq a_{n}, a_{1}+ a_{2}+ \cdots+ a_{n}= 300, a_{1}^{2}+ a_{2}^{2}+ \cdots+ a_{n}^{2}> 10000$$ With $n> 3,$ ...
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Ideals corresponding to principal genus

Given a discriminant $D<0$, $D\equiv 0, 1\bmod{4}$, there is a well-known bijection between primitive (positive definite) reduced forms of that discriminant and ideal classes in a particular order ...
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Optimization of a unique quadratic form

For the classic quadratic form $x^{\top}Ax$ ($x\in\mathbb{R}^n, A\in\mathbb{R}^{n \times n}$), we know that to solve the optimization problem $\max_{x}x^{\top}Ax \text{ subject to } ||x||_2=1$, we can ...
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Understanding a specific process of finding the derivative of $x^TAx$

I am referring to @copper.hat's response to : Derivative of Quadratic Form. I do not have the reputation to reply directly. My goal is to find a way to better differentiate and understand these ...
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Quadratic forms into sums of squares of linear forms

I would like to know whether there exists software or online calculators that turn quadratic forms directly into sums/differences of squares of linear forms? For instance: For $q(x,y,z) = 2x^2 - 2y^2 -...
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lower semi-bounded imply symmetric

A quadratic form is a map $q: Q(q) \times Q(q) \rightarrow \mathbb{C}$, where $Q(q)$ is a dense linear subset of the Hilbert space $H$. If $q(\phi,\psi)=\overline{q(\psi,\phi)}$, then we say q is ...
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Extremal positivity property of complete square

Consider the bilinear form $\displaystyle A(x,y) = \sum_{i, j} a_{i j} x_i y_j$ and biquadratic form $\displaystyle B(x,y) = \sum_{i, j, k, l} b_{i j k l} x_i y_j x_k y_l$ where $x, y \in \mathbb{R}^n$...
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Differentiate vector transpose using rules [duplicate]

I am referring to Tom Minka's Old and New Matrix Algebra Useful for Statistics. I don't have the book by Magnus & Neudecker so I can't refer to the details of the theory. Regarding rules (6): $d(...
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What is the gap(s) here ?? And why does there exist no such an $l\left ( a, b, c \right )\geq 0$

Given three real numbers $a, b, c,$ we have the following cyclic polynomial $H$ $$\left ( c- a \right )^{2}- \left ( a- b \right )\left ( b- c \right )= \left ( c+ a- 2b \right )^{2}+ {\color{Red} 3}\...
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Solving multivariate quadratic equations over the integers

I am looking for a method (if it exists) to solve over the integers the following sum of squares equation: $$ x_1^2 + x_2^2+x_3^2 + \cdots + x_n^2 = m,$$ with $m \in \mathbb{N}.$ Someone has any idea ...
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What is the point of representing a conic in homogeneous form?

In this textbook I am studying, they state the following: The equation of a conic in inhomogeneous coordinates is $$ax^2+bxy+cy^2+dx+ey+f=0$$ ie. a polynomial of degree 2. "Homogenizing" ...
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Prove that $\max_x \frac{(x^Tv)^2}{x^TBx}=v^TB^{-1}v$

Let $B>0$ be positive definite $d \times d$ matrix, and $v \in \mathbb R^d$. Prove that $$\max_x \frac{(x^Tv)^2}{x^TBx}=v^TB^{-1}v$$ Hint: Change of variable $z=B^{1/2}x$ $$\begin{split}\max_z\...
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Factorisation of a given polynomial into another with integer roots

Source: Challenge and Thrill of Pre-College Mathematics "Find all integers $a$ such that $$(x-a)(x-12)+2$$ can be factored into $(x-b)(x-c)$, such that $b$ and $c$ are integers." My attempt: ...
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Prove that all diagonal entries of a negative definite matrix are negative

We know that all principal minors of order one are nonpositive, but is there a way to prove that all of them are negative? I looked at a similar question here but the solution is too vague, so if ...
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122 views

Get wrong answer on $\frac{\partial \mathbf{x}^{\top} \mathbf{A} \mathbf{x}}{\partial \mathbf{x}}$ when using graph

I can use the product rule to obtain $\frac{\partial \mathbf{x}^{\top} \mathbf{A} \mathbf{x}}{\partial \mathbf{x}} = \mathbf{x}^{\top} \frac{\partial \mathbf{A} \mathbf{x}}{\partial \mathbf{x}}+(\...
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38 views

How is this value of a slope for a parabola

My teacher wrote that $$y + \frac{D}{4a} =a*(x+\frac {b}{2a})^2$$ is value of slope . I thought it could be instantaneous slope then since slope is of a line and not a parabola. Then , my question is ...
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39 views

Convergence of a quadratic form in probability

Suppose $x$ is a random vector in $R^n$ with positive definite covariance matrix, $R$. I'm interested in the convergence in probability of the quadratic form, $\dfrac{1}{n}x^TR^{-1}x$. If $x$ were ...
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Derivation help for $E[x^{\top}Ax] = tr(AP)$: missing the trace operator, using the moment generating function for Gaussian $x$ and gradient

Following notations are used: $x$: vector-valued random variable $x$ (with its mean $\bar{x}$ = 0). $P$: covariance of $x$ $tr(\cdot)$: trace operator $E[\cdot]$: expectation $M_x(s)$: Moment ...
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Monic Polynomial with absolute 1 as all coefficients

Determine all monic polynomials $p(x)$ with integer coefficients of degree two for which there exists a polynomial $q(x)$ with integer coefficients such that $p(x)q(x)$ is a polynomial having all ...
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A basis such that the bilinear form $\phi$ is diagonal, but the operator $T$ is not

Let $Q(x,y,z) = 2(xy+xz+yz) - (x^2+y^2+z^2)$ be a quadratic form in $\mathbb R^3$ and let $\phi$ be its associated bilinear form. Let also $T:\mathbb R^3 \rightarrow \mathbb R^3$ be a linear operator ...
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How to solve $x^2 - ny - 2 = 0$ for integer solutions, with different coefficients of $y$?

$$x^2 - ny - 2 = 0, n \in Z$$ I put this equation in wolframalpha, and got integer solutions for random n = 7, 23, 31, 47, 49, 71, 73, 343. for n = 7 it gave the integer solutions as $$x = 7 n + 3, y =...
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Upper and Lower Eigenvalue Bounds for $X^{T} A X$

Let $A \in \mathbb{R}_{+}^{n \times n}$ be diagonal with $i$th diagonal element $a_{ii}$ (i.e. all $a_{ii} > 0$), and let $X \in \mathbb{R}^{n \times p}$ be a matrix with rank $p$ where $n \ge p$ (...
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Determine if $f(x,y)=(1+\sin(x+y)) \ln(1+2x+y)-2x-y$ has a maximum at the origin

I want to determine if the function $f(x,y)=(1+\sin(x+y)) \ln(1+2x+y)-2x-y$ has a local extrema at the origin, and if so determine its characteristic. I found the quadratic form of the function ...
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32 views

Average of Two Quadratic Forms

I'm trying to show the following: $$\frac{1}{2}(x^{T}Ax +y^{T}Ay) \leq x^{T}Ay$$ with A symmetric and $||x||_{2}, ||y||_{2} \leq 1$. I'm not sure if it's true or not and was wondering if anyone had ...
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Quadratic form as sum of squares with fewer terms

Let $\ell_1,\dots,\ell_{m+n}$ be linear functionals over a finite vector space $V$. Suppose that $$Q(x)=\ell_1(x)^2+\dots+\ell_m(x)^2-\ell_{m+1}(x)^2-\dots-\ell_{m+n}(x)^2$$ is a quadratic form with ...
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Solve:$(1-y^2 +\frac{y^4}{x^2})p^2- (2\frac{y}{x})p + \frac{y^2}{x^2}=0$

I have the differential equation: $(1-y^2 +\frac{y^4}{x^2})p^2- (2\frac{y}{x})p + \frac{y^2}{x^2}=0$ where $p=\frac{dy}{dx}$ I have solved this up until $\frac{dy}{dx}$, but I am not able to reduce ...
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I can't wrap my head around it as no roots are given. [closed]

If $x^4+2x^3+px^2+qx+9=0$ is a complete square. $p$ and $q$ are positive. Find the value of $p$ and $q$.
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Line integral of a double dot product

Suppose one has a smooth vector field $\mathbf{f}$ in 3D. Given a closed contour $C$ which is a boundary of a surface $S$, a line integral can be calculated using the Stokes theorem: $$\oint\limits_{C}...
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1answer
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For a symmetric matrix $A$ and positive diagonal matrix $U$, is $U A + A U$ positive definite?

Let $A \in \mathbb{R^{n\times n}}$ be a positive definite matrix and $U \in \mathbb{R^{n\times n}}$ a diagonal matrix with positive entries, $A=A^\top \succ 0 $, $\:\:U=\textrm{diag}(u_1, \cdots, u_n)...
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37 views

Non-zero solutions to large systems of homogeneous quadratic equations

I'm looking at large systems of equations of the form $$\sum_{ij} M^{\alpha \beta}_{\lambda,ij} \bar{x}_{\alpha, i} x_{\beta, j} = 0 \quad \forall \alpha,\beta,\lambda.$$ Here, $x_{\alpha, i} \in \...
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Bound the contribution of superdiagonal block to the quadratic form $\mathbf{x}^T\mathbf{C}\mathbf{x}$

For $$ \mathbf{x} = \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \end{bmatrix}, \hspace{5mm} \mathbf{C} = \begin{bmatrix} \mathbf{C}_{11} & \mathbf{C}_{12} \\ \mathbf{C}_{21} & \mathbf{C}_{...
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1answer
81 views

Why are some functions called 'forms'?

The question is simple: why are some functions called 'forms'? Modular 'forms', bilinear 'forms', differential 'forms', quadratic 'forms', and so forth. It is not concretely a mathematical question ...
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1answer
35 views

Positive Definite Quadratic Form

I am following the solution of a problem which has the following matrix $$\begin{bmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 0 \end{bmatrix}$$ along with $dz = -dx - dy$ and also ...
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1answer
41 views

How to compute the class group?

I am not sure how to find the class group with a given discriminant d. First, we need to find the set of all primitive positive definite forms. Then we reduce all of them to split them into ...
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10 views

Relationship between class group and the set of reduced primitive forms.

I am confused with the definition of the class group. The definition of the class group is the set of equivalence classes of primitive binary quadratic forms of discriminant d. Hence it computes the ...
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13 views

Embedding a bivariate normal distribution in $\mathbb{R}^3$

Let $P$ be the precision matrix (inverse covariance matrix) of a bivariate normal distribution $p(x\in\mathbb{R}^2)\propto \exp(-\frac{1}{2}x^TPx)$. Further let $y=Ax$ where $A\in\mathbb{R}^{3\times2}$...
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25 views

Correctly factoring matrix in matrix quadratic equation

I am reading a paper where they make the following step. I do not understand how they got this. The paper is here: https://arxiv.org/pdf/1802.03050.pdf The step in question is on page 11 between ...
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1answer
29 views

Different representations of hyperbolic space

Two equivalent representations (not quite sure if that's the right word) of hyperbolic space $\mathcal{H}^n$ are the zero set of $- x_0^2 + x_1^2 + \ldots x_n^2 =-1 $ the quadratic space: vector ...
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3answers
35 views

theory of equation

let the roots of the equation $$x^4 -3x^3 +4x^2 -2x +1=0$$ be a , b, c,d then find the value of $$ (a+b) ^{-1} + (a+c) ^{-1}+ (a+d)^{-1} + (b+c)^{-1} + ( c+d)^{-1}+ (c+d)^{-1}$$ my solution i ...
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4answers
126 views

If a binary quadratic form primitively represents $n$ and $n^3$, must it be the identity form?

I've been searching numerically, and whenever a positive definite binary quadratic form can primitively represent an integer $n>1$ and $n^3$, it turns out to be equivalent to the identity form in ...

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