# Pick out which are diagonalisable:

Pick out which are diagonalisable:

1.any $n\times n$ unitary matrix over $\mathbb C$

2.any $n\times n$ hermitian matrix over $\mathbb C$

3.any $n\times n$ upper triangular matrix over $\mathbb C$

4.any $n\times n$ matrix over $\mathbb C$ having eigen values real

For 3 and 4 i think it is not diagonalisable since I can easily find matrices whose minimal polynomial does not split into distinct linear factors .I am not sure about 1 and 2

See this source for the spectral theorem that I'm working with here. The statement we want is that a matrix is normal (that is, $A^*A = AA^*$) if and only if it is unitarily diagonalizable (that is, there is a unitary $U$ such that $A = UDU^* = UDU^{-1}$, where $D$ is diagonal).