Derive a relation using Cauchy Schwarz inequality

So I am trying to derive the following relation using Cauchy Schwarz inequality for any collection of N real numbers:

This says that the square of the average is less or equal than the average of the squares.

So I want to use to derive this.

So needs to be therefore, I have as and as providing that:

{1}{N}" alt="{1}{N}" />

Am I going in the right direction? Can anyone provide me some guidance?

Re: Derive a relation using Cauchy Schwarz inequality

No. Currently, you have , so

Instead, try . Now, and are as you wanted, but , not . Fortunately, the inequality you are given has on the RHS, not .

Re: Derive a relation using Cauchy Schwarz inequality

I do apologise, it was a typo on my end and I forgot to add that I was using 1/N n amount of times.

But was my calculation of dot product incorrect then?

Re: Derive a relation using Cauchy Schwarz inequality

If you choose and , then those are real numbers. So, yes, your calculation of is incorrect, but your calculation of was correct. You would get . You cannot take it times because you chose to be a real number, not a number repeated times.

Re: Derive a relation using Cauchy Schwarz inequality

{ } and { }

Ok so I am still a little confused with whether I have this right or not now? And if this is correct for and then where do I go with , and in order to completely derive the inequality?

Re: Derive a relation using Cauchy Schwarz inequality

Yes, you have it right now.

Re: Derive a relation using Cauchy Schwarz inequality

Great. So now how do I show that the RHS is greater than or equal to the LHS using these facts?

Re: Derive a relation using Cauchy Schwarz inequality

To quote you:

Quote:

Originally Posted by

**MichaelH** So I want to use

to derive this.

Edit: Although, that should be , not .

Re: Derive a relation using Cauchy Schwarz inequality

Quote:

Originally Posted by

**SlipEternal** To quote you:

Edit: Although, that should be

, not

.

Ahh that's why I couldn't figure it out! Haha, my fault, should have double checked it. I was staring at = thinking... what am I supposed to do! Cheers mate.