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October 14th, 2018, 05:47 AM   #1
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Show this example is an inner product

\[B(x,y)=2\left( \sum_{i=1}^{n}x_{i}y_{i}\right)-\sum_{i=1}^{n-1}(x_{i}y_{i+1}+x_{i+1}y_{i})\]

I need help showing that this is positive definite as i think I have already shown it is symmetric and linear in the first variable. Thanks guys!
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October 15th, 2018, 09:32 AM   #2
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Are there restrictions on $x$ and $y$?

Fooling around with this using random numbers to fill $x$ and $y$, I was able to produce an example where $B(x,y) < 0$.

Actually, fooling with it a bit more, I see it's pretty common that $B(x,y)<0$.

It looks to occur approximately 10% of the time.

Last edited by skipjack; October 15th, 2018 at 04:08 PM.
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October 15th, 2018, 11:41 AM   #3
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Quote:
Originally Posted by romsek View Post
Are there restrictions on $x$ and $y$?

Fooling around with this using random numbers to fill $x$ and $y$, I was able to produce an example where $B(x,y) < 0$.

Actually, fooling with it a bit more, I see it's pretty common that $B(x,y)<0$.

It looks to occur approximately 10% of the time.
This isn't a problem. An inner product requires only that $B(x,x) \geq 0$ with equality only for $ x= 0$.
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Last edited by skipjack; October 15th, 2018 at 04:09 PM.
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October 15th, 2018, 01:58 PM   #4
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From Definition of Inner Product

B(x,y)=B(y,x)
B(ax+by,z)=aB(x,z)+bB(y,z)
B(x,x)>0 if x $\displaystyle \neq$ 0, and B(0,0)=0

Easiest with a summation convention if it can be set up that way. if I get a chance.
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October 16th, 2018, 05:46 AM   #5
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Hm yes I'll have a look later going to the gym to work on that chest right now!
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October 16th, 2018, 06:26 AM   #6
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Using summation convention over repeated indices:

$\displaystyle B(x,y)= 2x_{i}y_{i}-x_{i}y_{i+1}-y_{i}x_{i+1}$
$\displaystyle B(x,y)=B(y,x)$

$\displaystyle B(x,x)=2x_{i}x_{i}-2x_{i}x_{i+1}$

For n=3,
$\displaystyle B(x,x)= 2(x_{1}^{2}+x_{2}^{2} +x_{3}^{2})-2(x_{1}x_{2}+x_{2}x_{3})$
which is not always greater than zero and B(x,y) is not an inner product.

You can check second condition for IP as an exercise.
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October 16th, 2018, 07:13 AM   #7
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Originally Posted by zylo View Post
For n=3,
$\displaystyle B(x,x)= 2(x_{1}^{2}+x_{2}^{2} +x_{3}^{2})-2(x_{1}x_{2}+x_{2}x_{3})$
which is not always greater than zero and B(x,y) is not an inner product.
Of course $B(x,x)$ is not always positive - it is zero when $x$ is the zero vector. But it is positive whenever $x$ is not the zero vector, which is all we need.
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October 16th, 2018, 08:27 AM   #8
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Of course $B(x,x)$ is not always positive - it is zero when $x$ is the zero vector. But it is positive whenever $x$ is not the zero vector, which is all we need.
It didn't look like B(x,x) was always positive when I first did it. On closer inspection it seems to be true. Can you prove it?
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October 16th, 2018, 09:16 AM   #9
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Originally Posted by zylo View Post
It didn't look like B(x,x) was always positive when I first did it. On closer inspection it seems to be true. Can you prove it?
This amounts to showing $\sum_{i=1}^{n} x_i^2 > \sum_{i=1}^{n-1}x_i x_{i+1}$ whenever $x = (x_1,\dots,x_n) \neq 0$ for all $n \geq 2$. This is clearly true for $n = 2$ (if either of $x_1$ or $x_2$ is $0$, then $x_1^2 + x_2^2 > 0 = x_1 x_2$; else $x_1 x_2 \leq \operatorname{max}(x_1^2,x_2^2) < x_1^2 + x_2^2$).

Suppose it is true for some $n \geq 2$, and let $x = (x_1,\dots x_{n+1}) \neq 0$. Up to reordering the $x_i$, we may assume $x_{n+1} = \operatorname{max}(x_1,\dots,x_{n+1}$). Now either $(x_1, \dots, x_n) = 0$ (in which case the result is clearly true: we would have $\operatorname{LHS} = x_{n+1}^2 > 0 = x_n x_{n+1} = \operatorname{RHS}$) or

$$\sum_{i=1}^{n+1} x_i^2 - \sum_{i=1}^{n}x_i x_{i+1} = \underbrace{\left(\sum_{i=1}^{n} x_i^2 - \sum_{i=1}^{n-1}x_i x_{i+1}\right)}_{>0 ~ \text{by inductive hypothesis}} + \underbrace{\left(x_{n+1}^2 - x_n x_{n+1}\right)}_{\geq 0 ~ \text{as} ~ x_{n+1} ~ \geq ~ x_n} > 0,$$

which completes the induction.
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October 16th, 2018, 11:28 AM   #10
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Ref preceding post. It doesn't work. You can't change induction premise by reordering.

For example, suppose I have reordered to put $\displaystyle x_{2}$ in last place:
$\displaystyle B(x,x) = x_{1}^{2}+ x_{3}^{2}+ x_{2}^{2}-x_{1}x_{3}-x_{3}x_{2}$
Now put $\displaystyle x_{2}$ back where it was
$\displaystyle B(x,x) = x_{1}^{2}+ x_{2}^{2}+ x_{3}^{2}-x_{1}x_{2}-x_{2}x_{3}$
They aren't the same.

Last edited by zylo; October 16th, 2018 at 11:44 AM. Reason: clarify
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