1. (20 points) Find the eigenvalues and eigenvectors for each of the following matrices: a. 0...

 

Transcribed Image Text:

1. (20 points) Find the eigenvalues and eigenvectors for each of the following matrices: a. 0 1 0 GD ( ) 1 0 1 0 1 0/ b. A 10 x 10 matrix with zeroes everywhere except that all elements on the main diagonal have the value 2.0 and all the elements on the sub and superdiagonals have the value -1.0. C. Identify the five eigenvectors in part (b) belonging to the lowest five eigenvalues and, for each, plot a graph whose vertical coordinate is the component of the eigenvector and whose horizontal coordinate is the component number. Actually, in the underlying physical context, it would be more appropriate to plot graphs of the result of augmenting these eigenvectors by placing an element 0.0 both before the first element and after the last element in the eigenvector. If, for example, the eigenvectors are in the columns of evecs, then the statement >>plot([0.0; evecs(:,1); 0.0]) will plot the requested graph for the first column of evecs. Improve the appearance of the plot by adjusting the scales, adding labels, etc. You should see something close to the lowest several modes of a vibrating string fixed at both ends. d. Extra credit: Repeat (b) and (c) for a 100 x 100 matrix. 1. (20 points) Find the eigenvalues and eigenvectors for each of the following matrices: a. 0 1 0 GD ( ) 1 0 1 0 1 0/ b. A 10 x 10 matrix with zeroes everywhere except that all elements on the main diagonal have the value 2.0 and all the elements on the sub and superdiagonals have the value -1.0. C. Identify the five eigenvectors in part (b) belonging to the lowest five eigenvalues and, for each, plot a graph whose vertical coordinate is the component of the eigenvector and whose horizontal coordinate is the component number. Actually, in the underlying physical context, it would be more appropriate to plot graphs of the result of augmenting these eigenvectors by placing an element 0.0 both before the first element and after the last element in the eigenvector. If, for example, the eigenvectors are in the columns of evecs, then the statement >>plot([0.0; evecs(:,1); 0.0]) will plot the requested graph for the first column of evecs. Improve the appearance of the plot by adjusting the scales, adding labels, etc. You should see something close to the lowest several modes of a vibrating string fixed at both ends. d. Extra credit: Repeat (b) and (c) for a 100 x 100 matrix.