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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Matrix quadratic forms with non-symmetric shape parameter

A matrix quadratic form can be expressed as $\mathbf{Q}=\mathbf{XAX}^T$, where $\mathbf{X} \in \mathbb{R}^{n \times m}$ is a matrix and the shape parameter $\mathbf{A}\in \mathbb{R}^{m \times m}$ is a ...
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Hilbert symbol calculation

Is the next statement true? If $p$ is an odd prime, then $$(-1,-1)_p=1,$$ where $(\cdot ,\cdot)_p$ is the Hilbert Symbol. I can't related it with the definition of the Hilbert Symbol in which $$(a,b)...
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Isotropic positive definite quadratic forms over $\mathbb{Z}$

I don't know if this have any sense but, it is possible to have a positive definite $n$-ary quadratic form (over $\mathbb{Z}$ that is isotropic? Does the condition of positive definite made it ...
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Find the integer solution: $a+b+c=3d$, $\: a^{2} + b^{2} + c^{2}= 4d^{2}-2d+1$

Find the integer solutions: $$a+b+c=3d$$ $$a^{2} + b^{2} + c^{2}= 4d^{2}-2d+1$$ Attempt: Notice that $a=b=c=d=1$ is a solution. Other facts: Notice that $a^{2} + b^{2} + c^{2} > 0$, so $4d^{2}-...
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Sub-blocks of matrix quadratic equations

Suppose the following matrix quadratic equation has at least one real solution for $X$: $$ \begin{bmatrix} A & a \\ 0 & 1\\ \end{bmatrix} X^2 + \begin{bmatrix} B & b \\ 0 & 0\\ \...
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Primes representable by either $x^2+36y^2$ or $4x^2+9y^2$ [closed]

Is there a simple criterion for primes that are representable by either $x^2 + 36 y^2$ or $4x^2 + 9y^2$? This is not my area of expertise, so any pointers appreciated. I had a look in the Cox book &...
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Transforming singular quadratic forms

Let $F(x_1,\ldots,x_{r+1}) = \sum_{1\leq i,j \leq r+1}A_{ij}x_ix_j$ be an integral quadartic form with rank $r$ such that $A_{11} = 0$ and $A_{12} \neq 0$. Show that there exists a unimodular ...
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Show that $(x-a)^T(x-a)=\text{tr}(x_c, x_c)+n(a-\bar x)^2$

Show that $(x-a)^T(x-a)=\text{tr}(x_c, x_c)+n(a-\bar x)^2$ (Gentle Matrix Algebra exercise 3.2) This shows that the norm $\|x-a\|$ is minimized when $a=\bar x$. I went through the pages and got the ...
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Constant $SU(2)$ ASD connections on $\mathbb R^4$ are Flat

Let $A \in \mathfrak{su}(2) \otimes \mathbb R^4$, so $A$ is a collection of $4$ elements of $\mathfrak {su}(2), (A_0, \dots, A_3)$. We can consider the system of equations $$ [A_0, A_1] + [A_2, A_3] = ...
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$\overline{ \{\Delta > 0\} } = \{\Delta\ge 0\}$ for $\Delta$ a certain determinant function

Let $A=(a_{ij})$ be a real matrix. Consider the $(n+1)\times (n+1)$ bordered matrix $\tilde A= (a'_{ij})$ where $\tilde a_{ij} = a_{ij}$ for $1\le i,j\le n$, $a'_{ij} = 1- \delta_{ij}$ if $\max(i,j) =...
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A represents the quadratic form $q(x, y) = x^2 + 4xy + y^2 $; for what values of $(x, y) \in \mathbb{R}^2$ is $ q(x, y) =0$? [closed]

A represents the quadratic form $q(x, y) = x^2 + 4xy + y^2$ ; for what values of $(x, y) \in \mathbb{R}^2 $ is $ q(x, y) = 0$ ? Anyone here to help me, I would really appreciate
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Question regarding isometric relation on Pfister forms

I was going through this note on quadratic forms. While proving lemma 2.16, it is said that since $\pi_n$(a Pfister form) is hyperbolic and $\pi_n=\pi_{n-1}-\langle a_n\rangle\otimes\pi_{n-1}$, $\pi_{...
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Find values of the parameter for the given conic

Given $\mathbb{R}^2$ an affine space and the conics: $Q_\alpha:3x_1^2-\alpha x_1x_2+3x_2^2+14x_1-2x_2+3=0$ $C_\beta:x_1^2+2x_2^2+2\beta x_1x_2-6x_1+5=0$ $i)$ Find $\alpha$ and $\beta$ such that $Q_\...
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Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?

It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of rotating objects can be surprising (see the Dzhanibekov effect); in 4D it could be more surprising. A 2D or ...
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Finding a unit vector $v$ that makes only one quadratic form vanish

I was reading a proof on the non-convexity (even locally) of loss landscape in high-dimensional neural networks. Specifically, in the paper, it seems like the proof of proposition 2 at some point uses ...
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Meaningful upper bound on $\sum_{i=1}^n v_i^T \Big(\sum_{i=1}^n v_i v_i^T\Big)^{-1} v_i$

Let $v_1, \dots, v_n \in \mathbb{R}^d$ and $n \ge d$. Assume that the matrix $A$ is invertible. $$ A = \sum_{i=1}^n v_i v_i^T $$ Is it possible to simplify the expression $$\sum_{i=1}^n v_i^T A^{-1} ...
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What is the domain of $({\sqrt x})^2$?

I have the following question with me: Find the middle point of solution of the inequality.$$x^2+2(\sqrt x)^2-3\le0$$ I went through the following process: $$x^2+2x-3\le0$$ $$(x+3)(x-1)\le0$$ $$x\in[...
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Let $D,m$ be relatively prime integers with $m$ odd. Then $D \equiv 0,1 \pmod 4$ and $D \equiv b^2 \pmod m$ implies that $D \equiv b^2 \pmod{4m}$

This is from a proof in David A. Cox's Primes of the Form $x^2+ny^2$: Lemma 2.5 Let $D\equiv 0,1 \bmod 4$ be an integer and $m$ be an odd integer relatively prime to $D$. Then $m$ is properly ...
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Equivalent integral quadratic forms properly represent the same integers

Definitions: An integral quadratic form (IQF) is some instance of $f(x,y)=ax^2+bxy+cy^2$, where $a,b,c \in \mathbb{Z}$. Let $f(x,y),g(x,y)$ denote IQFs. We say $f(x,y)$ and $g(x,y)$ are properly ...
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Improper Equivalence of Integral Quadric Forms is not an Equivalence Relation

An integral quadric form is some instance of $f(x,y)=ax^2+bxy+cy^2$, with $a,b,c$ integers. Let $f(x,y),g(x,y)$ be two integral quadric forms. Then we say that they are improperly equivalent, denoted ...
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Positive definite matrix and Hormander Theory

Let $\varphi \in C_0^\infty$, $\varphi \neq 0$. We'll consider the inner product in $L^2.$ Let $\alpha ,\beta$ be multi-indices, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set$$\varphi _{...
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Over a finite field, which square matrices produce a zero quadratic form?

For which matrices $A \in (\mathbb{F}_p)^{n \times n}$ do we have $x^T A x=0$ for all $x \in (\mathbb{F}_p)^n$? Obviously, this is the case if $A=B-B^T$ for some $B$ (which is equivalent to saying ...
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A conditional negative definite quadratic form involving $\ln$ function

Let us consider the following property which is a constrained version of $(\star)$ (see Remark below): $$\begin{align*}\bbox[#EFF,15px,border:2px solid blue] {\begin{aligned}\text{For any n, for any} \...
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Classifying all binary quadratic forms over Z of a positive discriminant D

My textbook has a method for finding what quadratic forms are possible given a discriminant but that's only for positive definite binary quadratic forms. For example if the discriminant is $-100$ then ...
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Uniqueness of a Solution to a System of Quadratic Equations

Given $n$ positive semidefinite matrices $A_1,...,A_n \in \mathbb{R}^{d \times d}$, $x = 0$ is always a solution to the system of equations $$x^T A_1 x = \cdots = x^T A_n x.$$ When is this solution ...
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Quadratic forms: Existence of $(x,y)\in \mathbb{Z}^2 \setminus \{0\}$ such that $P(x,y) < 2\sqrt{\lvert \det(P) \rvert}$

I am stuck on the following exercise: Show that for any non-degenerate quadratic form $P$ over $\mathbb{R}$, that is either indefinite or positive definite, exists an integer point $(x,y) \in \mathbb{...
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Finding the maximum of quadratic forms

For some real vector $x \in \mathbb{R}^p$ and $p \times p$ positive definite real matrices $A_1, A_2, \dots, A _n$, consider another vector of quadratic forms: $$ (x^\top A_1 x, \dots, x^\top A_n x). $...
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Find the signature of a bilinear form given by a matrix

I'm trying to complete the bilinear form given by the matrix $$M=\left(\begin{array}{ccc}1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 1\end{array}\right)$$ into squares to find the ...
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What is the relationship between $AA^T$ and $A^TA$?

In the matrix $A = \begin{pmatrix}1 & 2 \\ 3 & 4 \\ 5 & 6\end{pmatrix}$ then eigenvalues of $A^TA$ are eigenvalues of $AA^T$, and $AA^T$ has an eigenvalue of 0. I've also experimented ...
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Restriction of a quadratic form with signature $(1, n-1)$.

I want to prove the following fact. Let $f$ be a nondegenerate symmetric bilinear function on a vector space $V$ with positive index of inertia equal to 1. And let $f(x,x) > 0$ for some vector $x \...
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Can the jacobi symbol be used for the statement "n is represented by some quadratic form of discriminant d iff 4n is a square mod d"

We've been using the above statement repeatedly in a number theory course, but to find all primes that are represented by a quadratic binary form of discriminant d, we've been using $$(\frac{d}{4p}) = ...
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real quadratic fields with 2 ramified primes

I have observed that when considering real quadratic fields with class number 1 and 2 having 2 ramified primes, the first ramified prime was (almost always, like 98% of the time) congruent to $3 \bmod ...
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Quadratic formula- calculator gives an answer but cannot do it manually

Embarrassing but I have this equation: If I use my calculator's quadratic mode I get the correct answer. However, if I try to do it manually I get a problem because 16 - 4 x 4 x 840.15 gives a ...
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isotropic vector-space of a quadratic form

Consider the quadratic form $q(x) = x_1x_2 + x_3x_4$. Find a subvector space $V$ of dimension 2 such that $\left.q\right|_V = 0$. What are the possible values for rank and signature of $(V,\left.q\...
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Line and plane tangent to a quadric surface

I'm stuck with what is seemingly a simple exercise or problem of linear algebra. Notice that the problem is purely geometrical in $\Bbb R^3$ (no derivatives, differential calculus). Let $\mathbf{x} = ...
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Prove every function $f \in SO(2)$ is a composition of two reflections. [duplicate]

Prove every function $f \in SO(2)$ is a composition of two reflections. Hint: use the matrix representation for the reflection on the line $ax+by=0$ I have no idea how to solve this or what I have to ...
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Matrix of a restriction of a quadratic form

Given a symmetric matrix $M$ and its associated quadratic form $Q : \mathbb{R}^n\times\mathbb{R}^n \rightarrow \mathbb{R}$ is there an obvious way to write down a matrix of the quadratic form $Q|_{U}$,...
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MGF of quadratic form $\mathbf{y}^\intercal \mathbf{A} \mathbf{y}$ where $y\sim N_p(\mathbf{\mu},\mathbf{\Sigma})$

Theorem 5.2b of Linear Models in Statistics by Rencher and Schaalje is If $\mathbf{y}$ is $N_p(\mathbf{\mu},\mathbf{\Sigma})$, then the moment generating function of $\mathbf{y}^\intercal\mathbf{A}\...
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Is the polarization of quadratic form bounded?

Suppose that $q: V\times V \to \mathbb{C}$ is a bounded quadratic form, define $$ \begin{equation} \tilde q(\phi,\psi) = \frac{1}{4} [q(\phi + \psi) -q(\phi - \psi) + iq(\phi + i\psi) - iq(\phi - i\...
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Showing the polarization of (complex) quadratic form is sesquilinear. [duplicate]

Let $q: V\times V \to \mathbb{C}$ on a complex vector space $V$ be a quadratic form. Define $\tilde q$ by the polarization identity: $$ \begin{equation} \tilde q(\phi,\psi) = \frac{1}{4} [q(\phi + \...
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Determine if the form $x_1^2 + x_2^2 -15(x_3^2+x_4^2)$ is isotropic over $\mathbb{Q}$.

I'm trying to show whether the form $x_1^2 + x_2^2 -15(x_3^2+x_4^2)$ is isotropic over $\mathbb{Q}$. I tried to apply the strong Hasse principle, but I don't get how to show isotropicity over $\mathbb{...
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If $f$ and $g$ two non-degenerate quadratic forms If $\delta(f)=\delta(g)$ is square in $\mathbb{F}_q$ then $f$ and $g$ are equivalent.

Let $f$ and $g$ two non-degenerate quadratic forms of a vecor space $E$ over a finite field $\mathbb{F}_q$ we note by $\delta(q)$ the discriminant of a quadratic form $q$. I want to show that if $\...
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Condition for two quadratic equation to have one common root (Simplification)

If a,b,c are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx2+2ex+f=0$ have a common root if $\frac da, \frac eb, \frac fc$ are in: Arithmetic Progression Geometric Progression ...
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To find Possible values of combinations of a & b for $a^2 + b^2 = 576$ or any quadratic equation under certain restrictions.

Q: Find possible values of combination of a&b such that $a^2 + b^2 = 576$ or any other quadratic equation under certain restrictions. The sole purpose is to understand how can we do it: CONDITION: ...
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Find an orthonormal basis and the signature of the quadratic form

Consider the quadratic form given by the matrix below (in the canonical basis) \begin{pmatrix} 1 & 1 & -1\\ 1 & 1 & 3\\ -1 & 3 & 1 \end{pmatrix} Find an orthonormal basis of it ...
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About another proof of Witt's theorem

This is a proof of Witt's cancellation theorem from Uzi Vishne's book (I wrote it in my words (in english) so if there is anything that does not seem accurate please tell me). Witt's cancellation ...
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Is every formally real field isomorphic to a subfield of the reals?

A formally real field is a field $K$ such that $-1$ is not a sum of squares in $K$. Clearly subfields of $\mathbb{R}$ are formally real. I also know finite fields and algebraically closed fields are ...
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2 votes
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Finding the equation of the Parabola

The parabola $y = x^2 + bx + c$ has the following properties: The point on the parabola closest to $(12,3)$ is the intersection with the $y$ axis of the parabola. The parabola passes through $(-5,0).$...
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How can I write $\alpha^2+\alpha{x}+\beta{x}+xy-\beta^2$ as a sum or difference of up to four squares?

I need to express this quadratic form $$\alpha^2+\alpha{x}+\beta{x}+xy-\beta^2$$ as a sum or difference of squares. (maximum 4 squares) How could I do that? It seem so difficult for me. I let you ...
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Quadratic form of a trace

Let $V := \left\{ X \in \mathfrak{gl}(2,\mathbb{R}) \mid \mbox{tr}(X) = 0 \right\}$ be a vector space over $\mathbb{R}$. Prove that function $$V\ni X \mapsto q(X) := \mbox{tr}(XDX^{T}),$$ where $$D=\...
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