In this chapter, we introduced normal matrices as those matrices that commute with their Hermitian conjugate. This seems like a weak requirement, but it's very easy to construct matrices for which this is not true.

(a) Consider a general, complex-valued \(2 \times 2\) matrix \(\mathbb{M}\) where

\[\mathbb{M}=\left(\begin{array}{ll}a & b \tag{3.148}\\c & d\end{array}\right)\]

for complex \ (a, b, c, d\). If \(\mathbb{M}\) is normal, what constraints does that place on the values of \ (a, b, c, d\) ?

(b) An upper-triangular matrix is a matrix that only has non-zero entries on the diagonal and above it; a lower-triangular matrix is defined analogously. For example, the \(2 \times 2\) matrix \[\left(\begin{array}{ll}a & b \tag{3.149}\\0 & d\end{array}\right)\] is upper triangular. Show that a general upper-triangular matrix of arbitrary dimension cannot be normal if there are non-zero entries above the diagonal.

For a matrix \(\mathbb{M}\), you just have to show that one element of \(\mathbb{M M}^{\dagger}\) does not equal the corresponding element of \(\mathbb{M}^{\dagger} \mathbb{M}\).