Symmetric Matrix Spectral Representation Exercises 2.4 2002/04/09 2002/08/05 Steven Cox cox@rice.edu Steven Cox cox@rice.edu Jacob Grabczewski jgrab@owlnet.rice.edu exercise spectral representation symmetric matrix A set of exercises after learning about the Spectral Representation, Gram-Schmidt Orthogonalization, and Diagonalization of a Symmetric Matrix. The stiffness matrix associated with the unstable swing is S 1 0 -1 0 0 1 0 0 -1 0 1 0 0 0 0 1 Find the three distinct eigenvalues, λ1 1 , λ2 2 , λ3 0 , along with their associated eigenvectors e 1 , 1 , e 1 , 2 , e 1 , 3 , e 3 , 1 , and projection matrices, P1 , P2 , P3 . What are the respective geometric multiplicities? Use the folk theorem that states "in order to transform a matrix it suffices to transform its eigenvalues" to arrive at a guess for S 12 . Show that your guess indeed satisfies S 12 S 12 S . Does your S 12 have anything in common with the element-wise square root of S? Show that P3 S . Assemble S+ 1 λ1 P1 1 λ2 P2 and check your result against pinv(S) in Matlab. Use S+ to solve S x f where f 0 1 0 2 and carefully draw before and after pictures of the unloaded and loaded swing. 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.