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Symmetric Matrix Spectral Representation Exercises

Module by: Steven Cox

Summary: A set of exercises after learning about the Spectral Representation, Gram-Schmidt Orthogonalization, and Diagonalization of a Symmetric Matrix.

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The stiffness matrix associated with the unstable swing is S=10-100100-10100001 S 1 0 -1 0 0 1 0 0 -1 0 1 0 0 0 0 1

  1. Find the three distinct eigenvalues, λ1=1 λ1 1 , λ2=2 λ2 2 , λ3=0 λ3 0 , along with their associated eigenvectors e 1 , 1 e 1 , 1 , e 1 , 2 e 1 , 2 , e 1 , 3 e 1 , 3 , e 3 , 1 e 3 , 1 , and projection matrices, P1 P1, P2 P2, P3 P3. What are the respective geometric multiplicities?
  2. Use the folk theorem that states "in order to transform a matrix it suffices to transform its eigenvalues" to arrive at a guess for S1/2 S 12 . Show that your guess indeed satisfies S1/2S1/2=S S 12 S 12 S . Does your S1/2 S 12 have anything in common with the element-wise square root of SS?
  3. Show that ℝP3=ℕS P3 S .
  4. Assemble S+=1λ1P1+1λ2P2 S+ 1 λ1 P1 1 λ2 P2 and check your result against pinv(S) in Matlab.
  5. Use S+S+ to solve Sx=f S x f where f=0102T f 0 1 0 2 and carefully draw before and after pictures of the unloaded and loaded swing.
  6. It can be very useful to sketch each of the eigenvectors in this fashion. In fact, a movie is the way to go. Please run the Matlab truss demo by typing truss and view all 12 of the movies. Please sketch the 4 eigenvectors of (1) by showing how they deform the swing.

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