Showing a matrix is normal and finding an orthonormal eigenbasis

Question

Given the matrix C:=$\frac{1}{2}\begin{pmatrix}1+i&1-i\\ 1-i&1+i\end{pmatrix}$

How would one

a) show C is normal

b)find an orthonormal eigenbasis for C

I'm not sure how to start this question, any help would be great!

Answer 1

A) You need to find the transpose conjugate, $C^H$ s.t $C^H$ has entries that are complex conjugates of $=C^T$ and $C^T$ is the transpose of $C$. If $CC^H=C^HC$ then $C$ is normal.

B) I think thi prhrase should help "If $A$ and $B$ are normal with $AB = BA$, then both $AB$ and $A + B$ are also normal. Furthermore there exists a unitary matrix U such that $UAU^∗ $ and $ UBU^∗$ are diagonal matrices. In other words $A $ and $ B$ are simultaneously diagonalizable.
In this special case, the columns of $ U^∗$ are eigenvectors of both $A$ and $B $ and form an orthonormal basis in $C^n$.
This follows by combining the theorems that, over an algebraically closed field, commuting matrices are simultaneously triangularizable and a normal matrix is diagonalizable – the added result is that these can both be done simultaneously."
If you need clarification, just ask. :-)