Question

Show that if $C \in \mathbb{C}^{p \times n}$ and $A \in \mathbb{C}^{n \times n}$, then $$ \operatorname{rank}\left[\begin{array}{l} C \\ C A \\ \vdots \\ C A^{\ell} \end{array}\right]=\operatorname{rank}\left[\begin{array}{l} C \\ C\left(\alpha I_n+A\right) \\ C\left(\alpha I_n+A\right)^{\ell} \end{array}\right] $$ for every positive integer $\ell$ and every point $\alpha \in \mathbb{C}$.

   Show that if $C \in \mathbb{C}^{p \times n}$ and $A \in \mathbb{C}^{n \times n}$, then
$$
\operatorname{rank}\left[\begin{array}{l}
C \\
C A \\
\vdots \\
C A^{\ell}
\end{array}\right]=\operatorname{rank}\left[\begin{array}{l}
C \\
C\left(\alpha I_n+A\right) \\
C\left(\alpha I_n+A\right)^{\ell}
\end{array}\right]
$$
for every positive integer $\ell$ and every point $\alpha \in \mathbb{C}$.

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Linear Algebra in Action

Linear Algebra in Action

Harry Dym 1st Edition

Chapter 19, Problem 20

Instant Answer

Step 1

We are given two matrices, \( C \in \mathbb{C}^{p \times n} \) and \( A \in \mathbb{C}^{n \times n} \), and we need to show that the rank of the matrix formed by stacking \( C, CA, \ldots, CA^\ell \) is the same as the rank of the matrix formed by stacking \( C, Show more…

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Show that if $C \in \mathbb{C}^{p \times n}$ and $A \in \mathbb{C}^{n \times n}$, then $$ \operatorname{rank}\left[\begin{array}{l} C \\ C A \\ \vdots \\ C A^{\ell} \end{array}\right]=\operatorname{rank}\left[\begin{array}{l} C \\ C\left(\alpha I_n+A\right) \\ C\left(\alpha I_n+A\right)^{\ell} \end{array}\right] $$ for every positive integer $\ell$ and every point $\alpha \in \mathbb{C}$.

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