Introduction - UCONN

THE MINIMAL POLYNOMIAL AND SOME APPLICATIONSKEITH easiest matrices to compute with are the diagonal ones. The sum and product ofdiagonal matrices can be computed componentwise along the main diagonal, and takingpowers of a diagonal matrix is simple too. All the complications of matrix operations aregone when working only with diagonal matrices. If a matrixAis not diagonal but can beconjugated to a diagonal matrix, sayD:=PAP 1is diagonal, thenA=P 1 DPsoAk=P 1 DkPfor all integersk, which reduces us to computations with a diagonal matrix.

THE MINIMAL POLYNOMIAL AND SOME APPLICATIONS 3 (See the proof of Theorem 2.2.) The converse is false: if all the eigenvalues of an operator are in F this does not necessarily mean the operator is diagonalizable.

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