MAT225 Section Summary: 5.3

Diagonalization

Summary

diagonalizable: A square matrix A is diagonalizable if A is similar to a diagonal matrix. That is, if for some diagonal matrix D.

The Diagonalization Theorem: is diagonalizable if and only if A has n linearly independent eigenvectors. Moreover, (where D is diagonal) if and only if the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are the eigenvalues.

Example: #2, p. 325

Rewrite the equation in the form AP=PD to understand what is going on! This is just the eigenvalue equation in partitioned form:

Theorem 6: An matrix with n distinct eigenvalues is diagonalizable.

Example: #10, p. 326

Theorem 7: Let A be an matrix whose distinct eigenvalues are .

1. For , the dimension of the eigenspace for is less than or equal to the multiplicity of the eigenvalue .
2. The matrix A is diagonalizable if and only if the sum of the dimensions of the distinct eigenspaces equals n.
3. If A is diagonalizable, and is a basis for the eigenspace corresponding to , then the collection of the bases forms an eigenvector basis for .

Example: #33, p. 326