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Fall 2018, problem 74

Prove that, for any real numbers a1,a2,,an, ni,j=1aiaji+j10.

Comments

Hubert
3 years ago

010(ni=1aixi1)2dx=10(ni=1aixi1.nj=1aixj1)dx

=10(ni,j=1aiajxi+i2dx)=ni,j=1aiaji+j1.

  • with equality for all zero -
Claudio
3 years ago

Hi, Hubert. What a magnificent solution!

After giving you a "+", I looked in vain for other demonstrations.

The only thing I found is a possible generalization of the result.

With the same demonstration we have:

For any real numbers a1,a2,,an and any real exponents b1,b2,...,bn with bj>12 it is: ni,j=1aiajbi+bj10

Sia
3 years ago

It's equivalent to prove Hilbert matrix is positive definite.