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.

A function $f : \mathbb{R^n}\to \mathbb{R^n}$ of the form $$f(x)=x^TAx=\sum_{i,j=1}^na_{ij}x_ix_j\qquad \text{where $A=(a_{ij})_{m \times n}$ be a real symmetric matrix and $x=x(x_1,x_2,...,x_n)\in \mathbb{R}$}$$ is called a quadratic form.

It is usually denoted by $Q(x)$.

Classification of the quadratic form $Q(x)=x^TAx$:

A quadratic form is said to be:

$a:\quad$ negative definite$: Q < 0$ when $x\neq 0$

$b:\quad$ negative semi definite$: Q ≤ 0$ for all $x$ and $Q = 0$ for some $x \neq 0$

$c:\quad$ positive definite$: Q > 0$ when $x \neq 0$

$d:\quad$ positive semi definite$: Q ≥ 0$ for all $x$ and $Q = 0$ for some $x \neq 0$

$e:\quad$ indefinite$: Q > 0$ for some $x$ and $Q < 0$ for some other $x$

Applications: The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers $\mathbb{Z}$ or the $p-$adic integers $\mathbb{Z}_p$. Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in $n$ variables has important applications to algebraic topology.

References:

"https://en.wikipedia.org/wiki/Quadratic_form"

"http://mathworld.wolfram.com/QuadraticForm.html"

2122 questions
64
votes
5 answers

How to take the gradient of the quadratic form?

It's stated that the gradient of: $$\frac{1}{2}x^TAx - b^Tx +c$$ is $$\frac{1}{2}A^Tx + \frac{1}{2}Ax - b$$ How do you grind out this equation? Or specifically, how do you get from $x^TAx$ to $A^Tx + Ax$?
43
votes
3 answers

Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = \det(A)\det(B) && \mathrm{tr}(A+B)= \mathrm{tr}(A)…
Mike F
  • 20,828
  • 3
  • 52
  • 98
33
votes
1 answer

Sum of squares of dependent Gaussian random variables

Ok, so the Chi-Squared distribution with $n$ degrees of freedom is the sum of the squares of $n$ independent Gaussian random variables. The trouble is, my Gaussian random variables are not independent. They do however all have zero mean and the same…
31
votes
5 answers

Derivative of Quadratic Form

For the Quadratic Form $X^TAX; X\in\mathbb{R}^n, A\in\mathbb{R}^{n \times n}$ (which simplifies to $\Sigma_{i=0}^n\Sigma_{j=0}^nA_{ij}x_ix_j$), I tried to take the derivative wrt. X ($\Delta_X X^TAX$) and ended up with the following: The $k^{th}$…
27
votes
7 answers

How to calculate the gradient of $x^T A x$?

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior derivative. The proof goes as follows: $ y =…
27
votes
2 answers

Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?

We did in class $x^2+y^2$, which was easy, and we had for homework $2x^2+2xy+3y^2$, which I did (its values (if not square) must be divisible by form primes, or of the form $x^2+5y^2$, and clearly they can't all be the latter!). Just to test my…
simplequestions
  • 673
  • 4
  • 11
25
votes
1 answer

Proof of Legendre's theorem on the ternary quadratic form

Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff $\left(\frac{-bc}{a}\right)=\left(\frac{-ac}{b}\right)=\left(\frac{ab}{c}\right)=1$. I'm trying to prove this…
quanta
  • 12,023
  • 3
  • 45
  • 84
23
votes
1 answer

Intuition/meaning behind quadratic forms

My professor just covered quadratic forms, but unfortunately did not give any intuition behind their meaning, so I'm hoping to get some of that cleared up. I know that we define a quadratic form as $Q(x) = x^T Ax$ , for some symmetric (i.e…
NNN
  • 1,732
  • 8
  • 27
22
votes
2 answers

What's a BETTER way to see the Gauss's composition law for binary quadratic forms?

There is a group structure of binary quadratic forms of given discriminant $d$: Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$ The composition of two binary quadratic forms is defined as: $$[f] [f']=[(A,B,C)],$$ where…
21
votes
2 answers

What is a form?

I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that it is to a ring what a vector is to a field. I…
20
votes
2 answers

Polarization: etymology question

The polarization identity expresses a symmetric bilinear form on a vector space in terms of its associated quadratic form: $$ \langle v,w\rangle = \frac{1}{2}(Q(v+w) - Q(v) - Q(w)), $$ where $Q(v) = \langle v,v\rangle$. More generally (over…
20
votes
5 answers

Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$

Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec x$, consisting of vectors with integer…
20
votes
4 answers

reference for linear algebra books that teach reverse Hermite method for symmetric matrices

May 11, 2019. Evidently the original method should be attributed to Lagrange in 1759. I got confused, Hermite is much more recent. January 13, 2016: book that does this mentioned in a question today, Linear Algebra Done Wrong by Sergei Treil. He…
Will Jagy
  • 130,445
  • 7
  • 132
  • 248
19
votes
4 answers

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication corresponds to map application. This provides a…
Jack M
  • 26,283
  • 6
  • 57
  • 113
19
votes
5 answers

Differentiate $f(x)=x^TAx$

Calculate the differential of the function $f: \Bbb R^n \to \Bbb R$ given by $$f(x) = x^T A x$$ with $A$ symmetric. Also, differentiate this function with respect to $x^T$. How exactly does this work in the case of vectors and matrices? Could…
1
2 3
99 100