Class Notes (838,885)
Canada (511,141)
Mathematics (2,859)
MAT224H1 (138)
Lecture

Diagonalization

3 Pages
154 Views
Unlock Document

Department
Mathematics
Course
MAT224H1
Professor
Martin, Burda
Semester
Winter

Description
Lecture 8 notes by Y. Burda 1 Diagonalization of operators, continued Last time we showed that eigenvectors corresponding to dierent eigenvalues are linearly independent. Now we will use it to get a criterion for when an operator is diagonalizable. Theorem. Let T : V V be a linear operator. Suppose that its charac- n1 nk teristic polynomial factors as p (T) = (x ) 1 ... (x k) . Then it is diagonalizable if and only if dimE = n ior every i, i.e. geometric i multiplicity coincides with algebraic multiplicity for all eigenvalues. Proof. For one direction we suppose that the operator T is diagonalizable and try to prove that dimE i = n .iSince T is diagonalizable, there is a 0 0 ... . basis v ,...,v in which its matrix is diagonal: 0 . 1 n . .. .. . . . 0 0 ... 0 Since we know the characteristic polynomial of this matrix, we get that n 1 of the entries on the diagonal are equal to ,
More Less

Related notes for MAT224H1

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit