Questions tagged [quadratic-forms]
By Gabriel Cooper •
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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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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 ... linear-algebra quadratic-forms generalized-eigenvector- 51
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 $$\sum_{i=1}^n v_i^T A^{-1} ... linear-algebra quadratic-forms- 2,169
What is the domain of $({\sqrt x})^2$?
I have the following question with me: Find the middle point of solution of the inequality.$$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[... algebra-precalculus inequality quadratic-forms- 13
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 ... modular-arithmetic algebraic-number-theory quadratic-forms- 165
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 ... gcd-and-lcm quadratic-forms coprime- 165
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 ... number-theory equivalence-relations quadratic-forms- 165
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$$\varphi _{... linear-algebra quadratic-forms differential-operators- 263
Over a finite field, which square matrices produce a zero quadratic form?
For which matrices $A \in (\mathbb{F}_p)^{n \times n}$ do we have $x^T A x=0$ for all $x \in (\mathbb{F}_p)^n$? Obviously, this is the case if $A=B-B^T$ for some $B$ (which is equivalent to saying ... linear-algebra matrices finite-fields quadratic-forms- 351
A conditional negative definite quadratic form involving $\ln$ function
Let us consider the following property which is a constrained version of $(\star)$ (see Remark below): $$\begin{align*}\bbox[#EFF,15px,border:2px solid blue] {\begin{aligned}\text{For any n, for any} \... logarithms quadratic-forms a.m.-g.m.-inequality- 65.9k
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 ... number-theory quadratic-forms- 161
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 ... linear-algebra quadratic-forms- 93
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{... number-theory quadratic-forms geometry-of-numbers- 2,502
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: $$ (x^\top A_1 x, \dots, x^\top A_n x). $... eigenvalues-eigenvectors quadratic-forms- 1
Find the signature of a bilinear form given by a matrix
I'm trying to complete the bilinear form given by the matrix $$M=\left(\begin{array}{ccc}1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 1\end{array}\right)$$ into squares to find the ... matrices quadratic-forms bilinear-form sums-of-squares- 337
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 ... linear-algebra quadratic-forms- 4,310
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