# Characteristic Polynomial The eigenvalues are also the roots of the charac1eris1ic poly110111ial...

Characteristic Polynomial The eigenvalues are also the roots of the charac1eris1ic poly110111ial of the matrix. The MATLAB function poly will compute the characteristic polynomial satisfied by the Illatrix [i .e., p(W) = OJ. However, this may not be the miniI11uI11 order polynomial satisfied by the matrix, which is called the minimal poly110111ial. Since the DFf matrix has repeated eigenvalues, it is necessary to analyze the matrix directly to get the minima l polynomial.

a. Generate an instance of the DFf matrix for a small value of

b. Suppose that the matrix is applied twice (i.e., take the DFr of the DFr):

Define a new matrix J = W1 and observe that many entries of J turn out to be equal to zero. Then it is possible to write a simple expression for yin terms of x. Why should J be called a "flip matrix"?

c. Suppose that the matrix W is applied four times:

What is z in terms of x? d. Use the results of the previous two parts to determine the minimal polynomial of W . Compare to the MATLAB function poly. Show that both polynomials have the same roots, even though the multiplicities are different. Use polyvalm to apply both polynomials to the matrix W and verify that it does indeed satisfy both .