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 Exercise [20.09]b 
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Joined: 26 Mar 2010, 04:39
Posts: 109
Post Exercise [20.09]b
Here is a much simpler solution

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16 Jul 2010, 06:56

Joined: 12 Jul 2010, 07:44
Posts: 154
Post Re: Exercise [20.09]b
I agree that Gandalf's solution is much, much more complicated than the question seems to warrant. In fact, he seems to be writing his own textbook!

However, I do think you need to also prove, in your solution, that positive definite matrices are invertible. This isn't hard, however.

A matrix is a linear mapping \mathbb{R}^n \rightarrow \mathbb{R}^n.
It's invertible provided that this mapping is 1-1.
Therefore, to show an inverse of a matrix M exists, you need to show there are no two (non-equal) vectors x, y such that Mx=My.
This is equivalent to showing that there is no vector z (=x-y) such that Mz=0.
But we know this is true, because "positive definite" implies zMz>0.
Therefore, all positive-definite matrices are invertible!

06 May 2012, 12:16
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