Classification of N x N Matrices:

2.11 Classification of N x N Matrices:

\tilde{A} = (\begin{matrix} a_{11} \dots a_{1 N} \\ ⋮ \\ a_{N 1} \dots a_{N N} \end{matrix})

$det (\tilde{A}) \neq 0 \Rightarrow$ regular matrix
$\tilde{A} = {\tilde{A}}^{⊤} \Rightarrow$ symmetric matrix $a_{j k} = a_{k j}$ example $(\begin{array}{ccc} 1 & 0 & 4 \\ 0 & 2 & 2 \\ 4 & 2 & 3 \end{array})$
$\tilde{A} = - {\tilde{A}}^{⊤} \Rightarrow$ anti-symmetric matrix $a_{j k} = - a_{k j}$ , in particular $a_{j j} = 0 \Rightarrow tr (\tilde{A}) = 0$
$\tilde{A} = {\tilde{A}}^{+} \Rightarrow$ self-adjoined matrix or Hermite-matrix , this means:
$\tilde{A} = \overset{―}{({\tilde{A}}^{⊤})}, a_{j k} = \overset{―}{a_{k j}}$
example: $\tilde{A} = (\begin{array}{ccc} 1 & 2 & - i \\ 2 & 2 & 1 - i \\ i & 1 + i & 3 \end{array})$ ,
if $\tilde{A}$ is real then (ii) and (iv) are equivalent. $\to$ diagonal elements of a Hermite-matrix are real, because of $a_{j j} = \overset{―}{a_{j j}}$ . The determinant is also real, i.e. $det (\tilde{A}) \in R$ if $\tilde{A}$ is a Hermite matrix.
$\tilde{A} = - {\tilde{A}}^{+} \Rightarrow$ anti-Hermite matrix ( $\to$ diagonal elements vanish) in particular $a_{j j} = 0 \Rightarrow tr (\tilde{A}) = 0$
${\tilde{A}}^{⊤} = {\tilde{A}}^{- 1} \Rightarrow$ $\tilde{A}$ is called orthogonal (real case) $\tilde{A} {\tilde{A}}^{⊤} = {\tilde{A}}^{⊤} \tilde{A} = \tilde{I}$ if $\tilde{A}$ is complex than ${\tilde{A}}^{+} = {\tilde{A}}^{- 1}$ means that A is ”unitary”. Properties of orthogonal matrices: $det (\tilde{A}) = \pm 1$ (follows from $det (\tilde{A}) = det ({\tilde{A}}^{⊤})$ and $det (\tilde{A} {\tilde{A}}^{⊤}) = det (\tilde{A}) det ({\tilde{A}}^{⊤})$ ) if $\tilde{A}$ and $\tilde{B}$ orthogonal, then $\tilde{A} \cdot \tilde{B} = \tilde{C}$ is also orthogonal.
Example: rotation around z-axis

$\begin{array}{rcl} \tilde{A} & = & (\begin{array}{ccc} \cos ϕ & \sin ϕ & 0 \\ - \sin ϕ & \cos ϕ & 0 \\ 0 & 0 & 1 \end{array}) \\ \to {\tilde{A}}^{⊤} & = & (\begin{array}{ccc} \cos ϕ & - \sin ϕ & 0 \\ \sin ϕ & \cos ϕ & 0 \\ 0 & 0 & 1 \end{array}) \\ \Rightarrow \tilde{A} {\tilde{A}}^{⊤} & = & (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}) since \cos^{2} ϕ + \sin^{2} ϕ = 1 \end{array}$

Test:

$det (\tilde{A}) = 1 | \binom{\cos ϕ}{- \sin ϕ} \binom{\sin ϕ}{\cos ϕ} | = + 1 o.k.$
diagonal Matrix:

$\tilde{A} = (\begin{array}{cccc} a_{11} & 0 & \dots & 0 \\ 0 & ⋱ & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & a_{N N} \end{array})$

$det (\tilde{A}) = a_{11} \cdot a_{22} \cdot \dots \cdot a_{N N}$ for diagonal matrix $\tilde{A}$