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# h11-4 - a where a ∈ R is a member of the Borel sigma...

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COT5615 Mathematics for Intelligent Systems Fall 2011 Home Work Assignment 4: Due Wednesday 11/30/11 before class 1. Prove that if A is a square matrix (i.e., n × n ) then the eigen values of A T A and AA T are the same. 2. Prove that if A is an n × n real symmetric matrix, then det( A ) = Q n i =1 λ i where λ i is the i th eigen value of A . 3. Prove that the set of all rational numbers (and therefore the set of all irrational numbers) is a member of the Borel σ -algebra on R . Prove that the set comprising of any singleton element:
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Unformatted text preview: { a } where a ∈ R is a member of the Borel sigma algebra. 4. Derive the KullbackLeibler divergence between the two 1-dimensional Gaussian density functions f 1 ( x ) = 1 √ 2 πσ 2 e-( x-μ 1 ) 2 2 σ 2 and f 2 ( x ) = 1 √ 2 πσ 2 e-( x-μ 2 ) 2 2 σ 2 Note that the Gaussians differ only in their means and not their variances. 1...
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