lower bound. WLOG, let ${\mathbf{{\ensuremath{K}}}}$ have ${l}-1$ fully shaded, linearly independent rows, and write it as $$\begin{aligned}
  {\mathbf{{\ensuremath{K}}}}&= \begin{bmatrix}
                    k_{11} & k_{12} & \cdots & k_{1{M}} \\
                    \vdots & \vdots & \cdots & \vdots \\
                    k_{({l}-1)1} & k_{({l}-1)2} & \cdots & k_{({l}-1){M}} 
                   \end{bmatrix}.
 \end{aligned}$$ Since the cyclic index is ${l}$, this implies that at least one eigenvalue, say ${{\lambda}}$, has ${l}$ Jordan blocks. Define indices $j_1, j_2, \dots, j_{{l}} \in \{1,2,\dots,{M}\}$ as the columns corresponding to the leading entries of the ${l}$ Jordan blocks corresponding to ${{\lambda}}$. WLOG, let $j_1 = 1$. Using ideas similar to the last proof, we can write the observability matrix as $$\begin{aligned}
  {\mathcal{O}}_{{\Upsilon}}
   &:= 
   \begin{bmatrix}
    {k}_{11}{{\lambda}}^{t_1}  & \cdots & {k}_{1j_{{l}}}{{\lambda}}^{t_1} & \cdots\\
    \vdots & \ddots &\vdots & \ddots\\
    {k}_{11}{{\lambda}}^{t_{{L}}}  & {k}_{1j_{{l}}}{{\lambda}}^{t_{{L}}} & \cdots\\
    \vdots & \ddots & \vdots & \ddots\\
    {k}_{({l}-1)1}{{\lambda}}^{t_1}  \cdots & {k}_{({l}-1)j_{{l}}}{{\lambda}}^{t_1} & \cdots\\
    \vdots & \ddots &\vdots & \ddots\\
    {k}_{({l}-1)1}{{\lambda}}^{t_{{L}}}  & \cdots & {k}_{({l}-1)j_{{l}}}{{\lambda}}^{t_{{L}}} & \cdots
   \end{bmatrix}.
 \end{aligned}$$ Define ${\boldsymbol{{{\lambda}}}}:= \begin{bmatrix}{{\lambda}}^{t_1} & {{\lambda}}^{t_2} & \cdots {{\lambda}}^{t_{{L}}}\end{bmatrix}^T$. Then the above matrix becomes $$\begin{aligned}
  {\mathcal{O}}_{{\Upsilon}}
   &:= 
   \begin{bmatrix}
    {k}_{11}{\boldsymbol{{{\lambda}}}}& \cdots & {k}_{1j_2}{\boldsymbol{{{\lambda}}}}& \cdots & {k}_{1j_{{l}}}{\boldsymbol{{{\lambda}}}}& \cdots\\
    \vdots & \ddots & \vdots & \ddots &\vdots & \ddots\\
    {k}_{({l}-1)1}{\boldsymbol{{{\lambda}}}}& \cdots & {k}_{({l}-1)j_2}{\boldsymbol{{{\lambda}}}}& \cdots & {k}_{({l}-1)j_{{l}}}{\boldsymbol{{{\lambda}}}}& \cdots
   \end{bmatrix}.
 