Question

Check that the mvf $F(\lambda)$ defined by formula (19.3) does not change if $C$ is replaced by $C S, A$ by $S^{-1} A S$ and $B$ by $S^{-1} B$ for some invertible matrix $S \in \mathbb{C}^{n \times n}$.

   Check that the mvf $F(\lambda)$ defined by formula (19.3) does not change if $C$ is replaced by $C S, A$ by $S^{-1} A S$ and $B$ by $S^{-1} B$ for some invertible matrix $S \in \mathbb{C}^{n \times n}$.

Linear Algebra in Action

Linear Algebra in Action

Harry Dym 1st Edition

Chapter 19, Problem 1

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Step 1

Assume $F(\lambda)$ is defined by the formula: \[ F(\lambda) = C (\lambda I - A)^{-1} B \] where $C$, $A$, and $B$ are matrices of appropriate dimensions, and $\lambda$ is a scalar. We are asked to check if $F(\lambda)$ remains unchanged under the transformations Show more…

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Check that the mvf $F(\lambda)$ defined by formula (19.3) does not change if $C$ is replaced by $C S, A$ by $S^{-1} A S$ and $B$ by $S^{-1} B$ for some invertible matrix $S \in \mathbb{C}^{n \times n}$.

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