# Math Help - Matrix eigenvalues

1. ## Matrix eigenvalues

If you are told that a square matrix M is diagonalisable, and it has only one eigenvalue which is c, then how do you show that M=cI, where I is the identity matrix?

Alternatively, and more generally, can you say that the sum of the eigenvalues of a diagonalisable square matrix M is the sum of the values along the main diagonal of M? How would you prove that?

2. I think I understand the second part. What about the 1st part?

3. Originally Posted by GrassyRoots
I think I understand the second part. What about the 1st part?
Proof: Applying the definition of diagonalizability: M is diagble $\implies \exists$ invertible P such that $P^{-1} M P =\Sigma$. Since all its eigen values are c, $\Sigma = cI$. But $P^{-1} M P = cI \implies M = P(cI) P^{-1} = cI$. Thus $\Sigma = cI \implies M = cI$ $\, \quad \blacksquare$