Chapter 6, Problem 12E

Consider the matrices
$A=\left(\begin{array}{cc}5& -3\\ 3& -5\end{array}\right)$ and $B=\left(\begin{array}{cc}5& -3\\ 3& 5\end{array}\right)$
Note that the two matrices are the same except for their (2, 2) entries.

1. Use MATLAB to compute the eigenvalues of A and B. Do they have the same type of eigen-values? The eigenvalues of the matrices are the roots of their characteristic polynomials. Use the following MATLAB commands to form the polynomials and plot their graphs on the same axis system:
   $\begin{array}{l}p=poly\left(A\right);\\ q=poly\left(B\right);\\ x=-8:0.1:8;\\ z=zeros\left(size\left(x\right)\right);\\ y=polyval\left(p,x\right);\\ w=polyval\left(q,x\right);\\ plot\left(x,y,x,w,x,z\right)\\ holdon\end{array}$ The hold on command is used so that subsequent plots in part (b) will be added to the current figure. How can you use the graph to estimate the eigenvalues of A? What does the graph tell you about the eigenvalues of B? Explain.
2. To see how the eigenvalues change as the $\left(2,2\right)$ entry changes, let us construct a matrix C with a variable $\left(2,2\right)$ entry. Set
   $t=sym\left(\text{'}t\text{'}\right)C=\left[5,-3;3,t-5\right]$
   As t goes from 0 to 10, the $\left(2,2\right)$ entries of these matrices go from -5 to 5. Use the following MATLAB commands to plot the graphs of the characteristic polynomials for the intermediate matrices corresponding to $t=1,2,...,9:.$

Which of these intermediate matrices have real eigenvalues and which have complex eigenvalues? The characteristic polynomial of the symbolic matrix C is a quadratic polynomial whose coefficients are functions of t. To find exactly where the eigenvalues change from real to complex, write thediscriminant of the quadratic as a function of t and then find its roots. One root should be in the interval (0, 10). Plug that value of t back into the matrix C and determine the eigenvalues of the matrix. Explain how these results correspond to your graph. Solve for the eigenvectors by hand. Is the matrix diagonalizable?