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 tex2html_wrap_inline194 for some diagonal matrix D.

The Diagonalization Theorem: tex2html_wrap_inline198 is diagonalizable if and only if A has n linearly independent eigenvectors. Moreover, tex2html_wrap_inline194 (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 tex2html_wrap_inline194 in the form AP=PD to understand what is going on! This is just the eigenvalue equation in partitioned form:

displaymath188

Theorem 6: An tex2html_wrap_inline220 matrix with n distinct eigenvalues is diagonalizable.

Example: #10, p. 326

Theorem 7: Let A be an tex2html_wrap_inline220 matrix whose distinct eigenvalues are tex2html_wrap_inline228 .

  1. For tex2html_wrap_inline230 , the dimension of the eigenspace for tex2html_wrap_inline232 is less than or equal to the multiplicity of the eigenvalue tex2html_wrap_inline232 .
  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 tex2html_wrap_inline242 is a basis for the eigenspace corresponding to tex2html_wrap_inline232 , then the collection of the bases tex2html_wrap_inline246 forms an eigenvector basis for tex2html_wrap_inline248 .

Example: #33, p. 326


LONG ANDREW E
Sat Jan 29 20:59:18 EST 2011