# Eigenvalues/orthogonality/diagonalization

• Apr 26th 2010, 07:14 AM
krtica
Eigenvalues/orthogonality/diagonalization
Are all the following statements true?

Any subspace of R^n has an orthogonal basis.
Every square matrix is diagonalizable.
0 vector is an eigenvector of every square matrix.
If B=P^(-1)AP then A and B have the same eigenvalues.
Every linear transformation has an eigenvalue.
• Apr 26th 2010, 09:38 AM
tonio
Quote:

Originally Posted by krtica
Are all the following statements true?

Any subspace of R^n has an orthogonal basis.
Every square matrix is diagonalizable.
0 vector is an eigenvector of every square matrix.
If B=P^(-1)AP then A and B have the same eigenvalues.
Every linear transformation has an eigenvalue.

Yes, no, yes, yes, it depends (in the definition field? Not necessarily; in some field extension? Yes, always)

Tonio
• Apr 26th 2010, 10:15 AM
krtica
Thank you! Can you please explain why the second statement is false?
• Apr 26th 2010, 11:44 AM
dwsmith
In this thread you will find 2 or 3 matrices that aren't diagonalizable.

http://www.mathhelpforum.com/math-he...ponential.html