hw4 - Homework 4 - MAE143B Spring 2010: due Thursday April...

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Homework 4 - MAE143B Spring 2010: due Thursday April 29 Question 1: Matrix exponential computations by hand [ Hint: Do your computations by hand and check them using matlab.] Consider the matrix A = 1 0 0 - 1 1 1 - 4 - 4 - 3 . Part i: Show that the characteristic polynomial of A is det( λI - A ) = ( s - 1)( s + 1) 2 . Show that for matrices J = 1 0 0 0 - 1 1 0 0 - 1 and V = 1 0 0 2 1 0 3 2 1 we have V A = JV. Accordingly, J is the Jordan canonical form of the matrix A . Part ii: Compute exp( At ) . Part iii: Use the adjugate matrix, adj ( sI - A ) (see Wikipedia page ), to compute, by hand, the 3 × 3 matrix ( sI - A ) - 1 = det( sI - A ) - 1 × adj ( sI - A ) . Then take the element-by element inverse Laplace transform of this matrix to show that exp( At ) = L - 1 ± ( sI - A ) - 1 ² . Question 2: Convolution and responses Consider the linear system studied in class ¨ x ( t ) + 2 ˙ x ( t ) + x ( t ) = u ( t ) . Part i:
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This note was uploaded on 05/12/2010 for the course MAE 143B taught by Professor Bitmead during the Winter '10 term at San Diego.

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hw4 - Homework 4 - MAE143B Spring 2010: due Thursday April...

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