Enter the matrix X defined
in the worksheet. Compute its eigenvalues, and record their multiplicities.
Let x denote the eigenvalue
of greatest multiplicity. What must be the dimension of the eigenspace
corresponding to x for X to be diagonalizable?
Find a basis for the eigenspace
corresponding to x. Is X diagonalizable or not?
If you have a built-in command to construct a matrix whose columns are supposed to be eigenvectors for the matrix X, use this command to construct a matrix R.
If R has columns that are eigenvectors, then R-1XR should be
a diagonal matrix. Test this. What do you deduce about the command you
used to construct R?
If you don't have such a
built-in command, construct a matrix whose columns are eigenvectors of X and compute R-1XR. Explain your result. Why did you get the result you did?
