I have three matrices and I am trying to determine if they represent a self-adjoint linear transformation in some basis on an inner product space. The matrices are:
$$A_1 = \left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{array} \right) A_2 = \left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right) A_3 = \left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 2 \end{array} \right) $$
Now for the first one I can see immediately that this only has a 2 dimensional eigenspace so cannot be self-adjoint. For the other two I am quite lost though.
I know that in an orthonormal basis the matrices should be the conjugate transpose of themselves, but here we have no idea what the basis/inner product us, so I'm not sure how we can tell.
Any help is very much appreciated!
1 Answer
You want to use the following fact:
A matrix $A$ represents a self-adjoint linear transformation in some basis if and only if the $A$ is real diagonalizable.
To prove the forward direction, notice that if $A$ represents a self-adjoint linear transformation, then $A = X B X^{-1}$, where $B$ is self-adjoint with respect to the standard inner product. But $B$ is real diagonalizable, and hence $A$ is.
For the reverse direction, if $A = X \Lambda X^{-1}$, where $\Lambda$ is a real diagonal matrix, then $\Lambda$ is self-adjoint in the standard basis.
