ABSTRACT
This chapter presents diagonalizable matrices and focuses on the fundamental question first addressed by Cauchy in 1826. A matrix is diagonalizable if and only if its minimal polynomial has no multiple zeros. Elementary row operations are replaced by annihilator polynomial computations, and the working matrix shrinks in rank, not size. The chapter shows Frobenius’ theorem: every complex matrix is similar to an upper triangular matrix. It utilizes transform plots to understand the geometric meaning of eigenvalues and eigenvectors, and then shows that the eigenvectors of a matrix are easily computed from its eigenvalues. The minimal polynomial carries very valuable information about a matrix. From this information, it is possible to find all eigenvalues and eigenvectors and to decide if the matrix is diagonalizable. There do exist Markov chains with transition matrices that are not diagonalizable, but the Perron-Forbenius theory still prevails and they cause no problems.
