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.

Quote:

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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.

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

Tonio

Thank you! Can you please explain why the second statement is false?

In this thread you will find 2 or 3 matrices that aren't diagonalizable.

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