# Diagonalizable matrix

• July 26th 2009, 03:55 PM
bmixon
Diagonalizable matrix
Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix.
• July 26th 2009, 04:57 PM
Random Variable
Let $A$ be a diagonalizable matrix having only one eigenvalue $\lambda$.

By the defintion of diagonalizability, there exists an nonsingular matrix $P$ such that $B = P^{-1}AP$ (where B is a diagonal matrix).

The diagonal entries of $B$ are the eigenvalues of $A$; therefore, $B = \lambda I$ (which is a scalar matrix).

This means that A is similar to $\lambda I$. But the only matrix similar to a scalar matrix is itself. So A is a scalar matrix.

I coincidentally just proved the last part in another thread about 20 minutes ago.