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Unformatted text preview: Math 260, Spring 2012 Jerry L. Kazdan Problem Set 7 Due: In class Thursday, March 1. Late papers will be accepted until 1:00 PM Friday . Unless otherwise stated use the standard Euclidean norm . 1. Say you have a matrix A ( t ) = (( a ij ( t ))) whose elements a ij ( t ) depend on a parameter t ∈ R . We say that A ( t ) is differentiable at t if the limit lim h → A ( t + h ) − A ( t ) h exists. We then call this limit A ′ ( t ). It is easy to show that A ( t ) is differentiable if and only if all of the elements a ij ( t ) are differentiable. a) Say A ( t ) and B ( t ) are n × n matrices that are differentiable and let C ( t ) := A ( t ) B ( t ) Show that C ( t ) is differentiable and give a formula for C ′ ( t ) in terms of A , A ′ , B , and B ′ b) If A ( t ) is differentiable and invertible, show that A − 1 ( t ) is differentiable and give a formula for dA − 1 ( t ) /dt in terms of A , A − 1 , and A ′ . [ Suggestion: Imitate the proof for a 1 × 1 matrix.] 2. In each of the following find u ( x,t )....
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This note was uploaded on 03/06/2012 for the course MATH 260 taught by Professor Staff during the Spring '12 term at UPenn.
- Spring '12