Questions Tagged [Quadratic-Forms]
Ask Question Quadratic Forms Are Homogeneous Quadratic (Degree Two) Polynomials in $N$ Variables. in the Cases of One, Two, and Three Variables They Are Called...
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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 ...
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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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