Proof : All matrices are diagonalisable

Question

1. All matrices can be decomposed into the sum of a hermitian and an anti hermitian matrix

2. All hermitian and anti hermitian are diagonalisable

3. Sum of diagonal matrices is also a diagonal matrix

4. All matrices are diagonalisable.

Where does this "proof" go wrong????

Answer 1

It goes wrong because statement 4 doesn't follow from statements 1, 2, and 3. As PhoemueX pointed out in the comments, there is a difference between diagonal and diagonalizable. All diagonal matrices are diagonalizable, but not every diagonalizable matrix is diagonal.

While statements 1 and 2 correctly show that any matrix can be written as the sum of two diagonalizable matrices, statement 3 does not allow us to conclude that this sum is also diagonalizable.

In order to use statements 1, 2, and 3 to conclude statement 4, statement 3 would need to be "the sum of two diagonalizable matrices is also diagonalizable". But of course, this isn't true.

For example $\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$ is diagonal (and thus diagonalizable) and $\begin{bmatrix}0 & 1 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix}0 & 1 \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}^{-1}$ is diagonalizable, but their sum $\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}+\begin{bmatrix}0 & 1 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix} $ is not diagonalizable.

Answer 2

Diagonalizing each of its two decomposition matrices doesn't create a diagonalization of the whole matrix. The invertible matrix you use to diagonalize the two matrices you decompose it into may be different from one another, and therefore don't result in a single invertible matrix that can diagonalize the whole thing.

Answer 3

The Hermitian and the anti-hermitian components of the matrix may both be diagonalisable, but the corresponding diagonal matrices may have different bases. The eigen-basis of the Hermitian component may be different from the eigen-basis of the anti-Hermitian component.

We do get a diagonal matrix by adding these two diagonalised matrices, but this sum has nothing to do with the original matrix we started with. This diagonal matrix is the sum of matrices belonging to two different basis, so it's a pretty meaningless matrix.