Unformatted text preview: ⊥ . Applying this result to A t gives Range( A t ) = [Nullsp( A )] ⊥ . Back to the SVD: If r = rank( A ), then σ r > 0 and σ r +1 = 0. If we write U = [ U 1 , U 2 ], and V = [ V 1 , V 2 ], where U 1 ∈ R m × r and V 1 ∈ R n × r , then (the columns of) U 1 form an orthonormal basis (O.B.) for Range( A ), U 2 an O.B. for Nullsp( A t ), V 1 an O.B. for Range( A t ), and V 2 an O.B. for Nullsp( A ). It’s all there in the SVD. And more. A matrix of rank s which best approximates A in the 2-norm is A s ≡ s X j =1 σ j u j v t j . This implies that the singular values tell us about how close A is to matrices of a given rank (e.g. how close to singular is this square matrix?), and helps us to quantify the uncertainties in our data....
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
- Fall '10