MAT224H1 Lecture : Diagonalization
Document Summary
Last time we showed that eigenvectors corresponding to di erent eigenvalues are linearly independent. Now we will use it to get a criterion for when an operator is diagonalizable. Let t : v v be a linear operator. Suppose that its charac- teristic polynomial factors as pt (x) = (x 1)n1 . Then it is diagonalizable if and only if dim e i = ni for every i, i. e. geometric multiplicity coincides with algebraic multiplicity for all eigenvalues. For one direction we suppose that the operator t is diagonalizable and try to prove that dim e i = ni. Since t is diagonalizable, there is a basis v1, . , vn in which its matrix is diagonal: Since we know the characteristic polynomial of this matrix, we get that n1 of the entries on the diagonal are equal to 1, n2 of the entries on the diagonal are equal to 2 and so on.
