Normal, Hermitian, and unitary matrices

• Commutative matrices

  Two square matrices ${\bf A}$ and ${\bf B}$ commute if ${\bf A}{\bf B}={\bf B}{\bf A}$.

  Obviously all diagonal matrices commute.

• Hermitian matrix

  A square matrix ${\bf A}$ is a Hermitian matrix if it is equal to its complex conjugate transpose ${\bf A}^*=\overline{\bf A}^T={\bf A}$. If a Hermitian matrix $\overline{\bf A}={\bf A}$ is real, it is a symmetric matrix, ${\bf A}^T={\bf A}$.

• Unitary matrix

  ${\bf A}$ is a unitary matrix if its conjugate transpose is equal to its inverse ${\bf A}^*={\bf A}^{-1}$, i.e., ${\bf A}^* {\bf A}={\bf I}$. When a unitary matrix $\overline{\bf A}={\bf A}$ is real, it becomes an orthogonal matrix, ${\bf A}^T={\bf A}^{-1}$.

  The column (or row) vectors ${\bf a}_1,\cdots,{\bf a}_n$ of a unitary matrix ${\bf A}=[{\bf a}_1,\cdots,{\bf a}_n]$ are orthonormal, i.e. they are both orthogonal and normalized:
  

  $$\langle{\bf a}_i, {\bf a}_j\rangle={\bf a}_i^T\overline{\bf... ... \begin{array}{ll} 1 & i=j 0 & i\neq j \end{array} \right.$$
  
  
  As we will see later, any Hermitian matrix ${\bf A}$ can be converted to a diagonal matrix ${\bf\Lambda}$ (or diagonalized) by a particular unitary matrix ${\bf\Phi}$:
  
  $${\bf\Phi}^*{\bf A}{\bf\Phi}={\bf\Lambda}=diag[\lambda_1,\cdots,\lambda_n]$$
  
  
  where ${\bf\Lambda}$ is a diagonal matrix, i.e., all its off diagonal elements are 0.

• Normal matrix

  A square matrix ${\bf A}$ is normal if it commutes with its conjugate transpose: ${\bf A}^*{\bf A}={\bf A}{\bf A}^*$. If ${\bf A}^*={\bf A}^T$ is real, then ${\bf A}^T{\bf A}={\bf A}{\bf A}^T$.

  Obviously unitary matrices ( ${\bf A}^*={\bf A}^{-1}$), Hermitian matrices ( ${\bf A}^*={\bf A}$), and skew-Hermitian matices ( ${\bf A}^*=-{\bf A}$) are all normal. But there exist normal matrices not belonging to any of these