I'm teaching myself linear control systems through various online materials and the book Linear Systems Theory and Design by Chen. I'm trying to understand controllability indices. Chen says that looking at the controllability matrix $C$ as follows
$$ C = \begin{bmatrix} B & AB & A^2 B & \cdots & A^n B \end{bmatrix} = \begin{bmatrix} b_1 & \cdots & b_p & | & A b_1 & \cdots & A b_p & | & \cdots & | & A^{n-1} b_1 & \cdots & A^{n-1} b_p \end{bmatrix}$$
that reading left to right, the columns are linearly independent until some column $A^ib_m$, which is dependent on the columns to the left - this makes sense - but that subsequent columns associated with $b_m$ (e.g. $A^{i+k}b_m$) are also linearly dependent on their proceeding columns, or that "once a column associated with $b_m$ becomes linearly dependent, then all columns associated with $b_m$ thereafter are linearly dependent". Unfortunately this last point isn't intuitive to me at all - if anyone can shed some light on this or help me see it I'd be very grateful!
Thanks in advance for any help.
Luke
Say $A b_1$ is linearly dependent on the $b_i$, namely $A b_1 = \lambda_i b_i$ (summation implicit). Then $A^2 b_1 = A \lambda_i b_i = \lambda_i A b_i$, so $A^2 b_1$ depends linearly on the columns $A b_i$ which are on its left.