Two coupled channels of a linear amplifier transform an input amplitude pair into an…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Numerical and computational mathematics

Error, Chance and the Machine, Ch Rounding, backward error and the sensitivity of a computation

Problem

Two coupled channels of a linear amplifier transform an input amplitude pair into an output pair through

\mathsf{A} = \begin{pmatrix} 3 & 60 \\ 0 & 3\end{pmatrix}.

((a)) State the eigenvalues of

\mathsf{A}

and their ratio. ((b)) Compute

\|\mathsf A\|_2

and

\|\mathsf A^{-1}\|_2

, using

\|\mathsf A\|_2 = \sqrt{\lambda_{\max}(\mathsf A^{\rm T}\mathsf A)}

. ((c)) Compute

\kappa_2(\mathsf A)

and say whether it agrees with the eigenvalue ratio found in part (a).

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