Observability and Mathematics

6 Observability and operations on matrices

Let us take two matrices A and B,

$\begin{array}{c}\mathit{A}=\mathrm{(}{\mathit{a}}_{\mathit{i}\mathit{j}}\mathrm{)}\mathrm{,}\mathit{B}=\mathrm{(}{\mathit{b}}_{\mathit{i}\mathit{j}}\mathrm{)}\mathrm{,}\mathit{i}\in \{1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{k}\}\mathrm{,}\mathit{j}\in \{1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{m}\}.\end{array}$

We define the sum of these matrices from a ${\mathit{W}}_{\mathit{n}}$-observer point of view as

$\begin{array}{c}\mathit{A}{+}_{\mathit{n}}\mathit{B}=\mathrm{(}{\mathit{a}}_{\mathit{i}\mathit{j}}\mathrm{)}{+}_{\mathit{n}}\mathrm{(}{\mathit{b}}_{\mathit{i}\mathit{j}}\mathrm{)}=\mathrm{(}{\mathit{a}}_{\mathit{i}\mathit{j}}{+}_{\mathit{n}}{\mathit{b}}_{\mathit{i}\mathit{j}}\mathrm{)}.\end{array}$

We assume that all elements of this equality belong to ${\mathit{W}}_{\mathit{n}}$ or $\mathit{C}{\mathit{W}}_{\mathit{n}}$.

If

$\begin{array}{c}\mathit{A}=\mathrm{(}{\mathit{a}}_{\mathit{i}\mathit{j}}\mathrm{)}\mathrm{,}\mathit{B}=\mathrm{(}{\mathit{b}}_{\mathit{p}\mathit{q}}\mathrm{)}\mathrm{,}\mathit{i}\in \{1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{k}\}\mathrm{,}\mathit{j}\in \{1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{m}\}\mathrm{,}\mathit{p}\in \{1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{r}\}\mathrm{,}\mathit{q}\in \{1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{s}\}\mathrm{,}\end{array}$

and $\mathit{m}=\mathit{r}$, then we define the product of these matrices from a ${\mathit{W}}_{\mathit{n}}$-observer point of view as

$\begin{array}{c}\mathit{A}{\times }_{\mathit{n}}\mathit{B}=\mathit{D}=\mathrm{(}{\mathit{d}}_{\mathit{i}\mathit{q}}=\sum _{\mathit{j}=1}^{\mathit{m}}{\mathit{n}}^{}{\mathit{a}}_{\mathit{i}\mathit{j}}{\times }_{\mathit{n}}{\mathit{b}}_{\mathit{j}\mathit{q}}\mathrm{)}.\end{array}$

We assume that all elements of this equality belong to ${\mathit{W}}_{\mathit{n}}$ or $\mathit{C}{\mathit{W}}_{\mathit{n}}$.

And we write

$\begin{array}{c}\mathrm{\left(}\mathrm{\right(}\dots \mathrm{(}{\mathit{f}}_1{+}_{\mathit{n}}{\mathit{f}}_2\mathrm{)}\dots \mathrm{\left)}{+}_{\mathit{n}}{\mathit{f}}_{\mathit{N}}\mathrm{\right)}=\sum _{\mathit{j}=1}^{\mathit{m}}{\mathit{n}}^{}{\mathit{f}}_{\mathit{j}}\end{array}$

for ${\mathit{f}}_1\mathrm{,}\dots \mathrm{,}{\mathit{f}}_{\mathit{N}}$ if and only if the contents of any parenthesis belong to ${\mathit{W}}_{\mathit{n}}$ or $\mathit{C}{\mathit{W}}_{\mathit{n}}$.

We define the product of a scalar and a matrix from a ${\mathit{W}}_{\mathit{n}}$-observer point ...