October 16th, 2018, 12:53 PM  #11  
Senior Member Joined: Sep 2016 From: USA Posts: 503 Thanks: 281 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
 
October 16th, 2018, 05:16 PM  #12  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
Suppose the result holds for some $n \geq 2$. Say $(x_1, \dots, x_{n+1}) \neq 0$ and suppose $x_j$ is maximal amongst the $x_i$. Consider the $n$tuple $x = (x_1, \dots, \overline{x_j}, \dots, x_{n+1})$, where the overline indicates we are deleting the $j$th entry. If $x = 0$ then we're fine: $\operatorname{LHS} = x_j^2 > 0 = \operatorname{RHS}$, so suppose $x \neq 0$. If $j = n+1$, then my old argument works. Else $$\sum_{i=1}^{n+1} x_i^2  \sum_{i=1}^{n}x_i x_{i+1} = \underbrace{\left(\sum_{\substack{i=1 \\ i \neq j}}^{n+1} x_i^2  \sum_{\substack{i=1 \\ i \neq j}}^{n}x_i x_{i+1}\right)}_{>0 ~ \text{by inductive hypothesis applied to $x$}} + \underbrace{\left(x_{j}^2  x_j x_{j+1}\right)}_{\geq 0 ~ \text{as} ~ x_j ~ \geq ~ x_{j+1}} > 0.$$  
October 17th, 2018, 06:50 AM  #13 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
Ref previous post. Same problem. $\displaystyle B(x,x)=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}x_{1}x_{2}x_{2}x_{3}$ $\displaystyle B(x,x)=x_{1}^{2}+x_{3}^{2}x_{2}x_{3} +(x_{2}^{2}x_{1}x_{2})$ But you need $\displaystyle x_{1}x_{3}$ for induction to work. I don't think you can use induction on size of a vector space. 
October 17th, 2018, 07:20 AM  #14 
Senior Member Joined: Sep 2016 From: USA Posts: 503 Thanks: 281 Math Focus: Dynamical systems, analytic function theory, numerics  Your argument was fine the first time. The induction assumption is that it holds for any collection of $n$ real numbers. If you now give me a collection of $n+1$, induction can be applied to any $n$sized subset. Namely, you can assume the odd one out is the largest, smallest, or any other ordering you like.

October 17th, 2018, 07:49 AM  #15 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Apologies. Not really sure what I was thinking yesterday. For any $x = (x_1, \dots, x_n)$, we have $$\begin{align*} B(x,x) &= 2 \sum_{i=1}^{n}x_{i}^22\sum_{i=1}^{n1}x_{i}x_{i+1} \\ &= x_1^2 + x_n^2 + \sum_{i=1}^{n1} (x_i^2 + x_{i+1}^2  2x_i x_{i+1}) \\ &= x_1^2 + x_n^2 + \sum_{i=1}^{n1} (x_i  x_{i+1})^2 \\ &\geq 0 \end{align*}$$ Moreover, this shows $B(x,x) = 0$ if and only if $0 = x_1^2 = x_n^2 = (x_ix_{i+1})^2$ for all $i = 1, \dots, n1$. In other words, if and only if all the $x_i$ are equal to each other and $x_1 = x_n = 0$. But this is equivalent to $x = 0$. You certainly can use induction on dimension to prove things about (finitedimensional) vector spaces. It just didn't turn out to be helpful in this particular example, at least the way I was trying it. 
October 17th, 2018, 07:53 AM  #16 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
B is a quadratic form which can be written as x'Ax which is positive definite if the eigenvalues of A are positive. EDIT: Discovered last post when I posted this. It is correct and nice thanks. Do you have an example of induction on size of a vector space? Offhand I can only think of induction in a vector space, like associativity. Last edited by zylo; October 17th, 2018 at 08:17 AM. 
October 17th, 2018, 08:03 AM  #17  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
e.g. say $x = (x_1, x_2, x_3)$ and say $x_2 \geq x_1, x_3$. I want to show that $x_1^2 + x_2^2 + x_3^2  x_1 x_2  x_2 x_3 > 0$. Applying the inductive hypothesis to $(x_1, x_3)$ (assuming it's not $(0,0)$) tells me that $x_1^2 + x_3^2  x_1 x_3 > 0$. But now I'm stuck with this $x_1 x_3$ term... I can still conclude $x_1^2 + x_2^2 + x_3^2  x_1 x_3  x_2 x_3 > 0$ and $x_1^2 + x_2^2 + x_3^2  x_1 x_3  x_1 x_2 > 0$, but neither of these facts are of any use. This is used all the time in field theory. For instance, if you look at any book on elementary Galois theory, you'll see plenty of examples. Last edited by cjem; October 17th, 2018 at 08:37 AM.  
October 17th, 2018, 08:37 AM  #18 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  
October 17th, 2018, 08:54 AM  #19 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  
October 17th, 2018, 09:10 AM  #20 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Sure, but there are a lot of examples of what you want in field theory. Anyway, another option: proofs of Engel's and Lie's theorems (in the theory of Lie algebras). The standard proofs are quite explicitly done by induction on the dimension of vector spaces. Even if you don't understand all the details, the overall structure of the proof and the way induction is used should hopefully be clear.
Last edited by cjem; October 17th, 2018 at 09:19 AM. 

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