Define system $\backslash$Sigma $\_1$:$\{($&x $\_1=$A$\_1$ x$\_1+$B$\_1$ u$\_1$@&y$\_1=$C$\_1$ x$\_1),\backslash$Sigma $\_2$: $\{($&x $\_2=$A$\_2$ x$\_2+$B$\_2$ u$\_2$@&y$\_2=$C$\_2$ x$\_2).\backslash$Sigma $\_1,\backslash$Sigma $\_2$ are duality systems:

$(1)$ Depict the relationships betweenA$\_1,$B$\_1,$C$\_1$ and A$\_2,$B$\_2,$C$\_2$

$(2)$ Prove that the transfer function matrices of $\backslash$Sigma $\_1$ and $\backslash$Sigma $\_2$ are mutually inverted.systems:

$(1)$ Depict the relationships between $A_1,B_1,C_1$ and $A_2,B_2,C_2$

$(2)$ Prove that the transfer function matrices of ${}_1$ and ${}_2$ are mutually inverted.