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301_F10_hw6

# 301_F10_hw6 - AME 301 Homework#6 | Due Fri 1 Problem Set...

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AME 301 Homework #6 | Due Fri Oct 15, 2010 1. Problem Set 8.1 #18 Find the eigenvalues and eigenvectors. Determine the algebraic and geometric multiplicity for the repeated eigenvalue. 2. A linear elastic deformation is described by the matrix equation y = Ax , where x is the location of a point prior to the deformation and y is the location after the deformation. Consider the case A = 0 : 8 0 : 6 0 : 6 1 : 7 : (a) Determine the angles 1 and 2 for the principal directions of the deformation (in which y is parallel to x ). Are the principal directions orthogonal to each other? Could this have been predicted from the form of the matrix A ? (b) Find the expansion or contraction ratios j y j = j x j in the principal directions. 3. Problem Set 8.3 #14 Find the eigenvalues. Verify that the eigenvalues satisfy The- orem 1 on page 346 of Kreyszig (9th edition). 4. Consider the di erence equation x k +1 = Ax k , where A = 1 = 2 1 2 5 = 2 and x 0 = a b : (a) By direct matrix multiplication, obtain symbolic expressions for x 1 , x 2 and x k in terms of powers of A . (You do not need to obtain numerical values for the matrix

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301_F10_hw6 - AME 301 Homework#6 | Due Fri 1 Problem Set...

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