{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw4 - Homework 4 MAE143B Spring 2010 due Thursday April 29...

This preview shows pages 1–2. Sign up to view the full content.

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 . Question 2: Convolution and responses Consider the linear system studied in class ¨ x ( t ) + 2 ˙ x ( t ) + x ( t ) = u ( t ) . Part i: Show that the Laplace transform of the zero-input response when the initial conditions are x 0 and ˙ x 0 is given by X zi ( s ) = sx 0 + 2 x 0 + ˙ x 0 ( s + 1) 2 = x 0 s + 1 + x 0 + ˙ x 0 ( s + 1) 2 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

hw4 - Homework 4 MAE143B Spring 2010 due Thursday April 29...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online