Observability and Mathematics

8 Observability and matrix Lie algebras

Let us consider now the matrix Lie algebras from a ${\mathit{W}}_{\mathit{n}}$-observer point of view. For reference, see the book [5].

In classic algebra, we call the finite-dimensional linear space K over the field of real ($\mathbb{R}$) or complex ($\mathbb{C}$) numbers a finite-dimensional Lie algebra over the field of real ($\mathbb{R}$) or complex ($\mathbb{C}$) numbers if for any two elements $\mathit{a}\mathrm{,}\mathit{b}\in \mathit{K}$ there is an additionally defined third element $\mathit{d}\in \mathit{K}$, and we call this third element the commutator of first two and write

$\begin{array}{c}[\mathit{a}\mathrm{,}\mathit{b}]=\mathit{d}\mathrm{,}\end{array}$

such that the following conditions are satisfied:

1.

(Linearity by summation and scalar multiplication)

$\begin{array}{c}[\mathit{\alpha }\mathit{a}+{\mathit{\alpha }}^{\prime }{\mathit{a}}^{\prime }\mathrm{,}\mathit{b}]=\mathit{\alpha }[\mathit{a}\mathrm{,}\mathit{b}]+{\mathit{\alpha }}^{\prime }[{\mathit{a}}^{\prime }\mathrm{,}\mathit{b}]\end{array}$

for any $\mathit{a}\mathrm{,}{\mathit{a}}^{\prime }\mathrm{,}\mathit{b}\in \mathit{K}$ and $\mathit{\alpha }\mathrm{,}{\mathit{\alpha }}^{\prime }\in \mathbb{R}$ or $\mathit{\alpha }\mathrm{,}{\mathit{\alpha }}^{\prime }\in \mathbb{C}$.

2.

(Commutator antisymmetry)

$\begin{array}{c}[\mathit{a}\mathrm{,}\mathit{b}]=-[\mathit{b}\mathrm{,}\mathit{a}]\end{array}$

for any ...