Let (C, A) ∈ℂ^p ×n ×ℂ^n ×n be an observable pair and let 𝐮(λ)=𝐮0+λ𝐮1+⋯+λ^k 𝐮k be a vector polyno

Step 1: **Understand the problem and definitions**. We need to show that the function $C(lambda

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Let $(C, A) \in \mathbb{C}^{p \times n} \times \mathbb{C}^{n \times n}$ be an observable pair and let $\mathbf{u}(\lambda)=\mathbf{u}_0+\lambda \mathbf{u}_1+\cdots+\lambda^k \mathbf{u}_k$ be a vector polynomial with coefficients in $\mathbb{C}^n$. Show that $C\left(\lambda I_n-A\right)^{-1} \mathbf{u}(\lambda)$ has a pole at $\alpha$ if and only if $\left(\lambda I_n-A\right)^{-1} \mathbf{u}(\lambda)$ has a pole at $\alpha$. [HINT: Try Exercise 19.16 first to warm up.]

   Let $(C, A) \in \mathbb{C}^{p \times n} \times \mathbb{C}^{n \times n}$ be an observable pair and let $\mathbf{u}(\lambda)=\mathbf{u}_0+\lambda \mathbf{u}_1+\cdots+\lambda^k \mathbf{u}_k$ be a vector polynomial with coefficients in $\mathbb{C}^n$. Show that $C\left(\lambda I_n-A\right)^{-1} \mathbf{u}(\lambda)$ has a pole at $\alpha$ if and only if $\left(\lambda I_n-A\right)^{-1} \mathbf{u}(\lambda)$ has a pole at $\alpha$. [HINT: Try Exercise 19.16 first to warm up.]

Linear Algebra in Action

Harry Dym 1st Edition

Chapter 19, Problem 17

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Step 1

We need to show that the function $C(\lambda I_n - A)^{-1} \mathbf{u}(\lambda)$ has a pole at $\lambda = \alpha$ if and only if $(\lambda I_n - A)^{-1} \mathbf{u}(\lambda)$ has a pole at $\lambda = \alpha$. Recall that a function $f(\lambda)$ has a pole at Show more…

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Let $(C, A) \in \mathbb{C}^{p \times n} \times \mathbb{C}^{n \times n}$ be an observable pair and let $\mathbf{u}(\lambda)=\mathbf{u}_0+\lambda \mathbf{u}_1+\cdots+\lambda^k \mathbf{u}_k$ be a vector polynomial with coefficients in $\mathbb{C}^n$. Show that $C\left(\lambda I_n-A\right)^{-1} \mathbf{u}(\lambda)$ has a pole at $\alpha$ if and only if $\left(\lambda I_n-A\right)^{-1} \mathbf{u}(\lambda)$ has a pole at $\alpha$. [HINT: Try Exercise 19.16 first to warm up.]

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