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Exercise 1.6.21

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$A^{k} = X Λ^{k} X^{- 1}$ approaches the zero matrix as $k \to \infty$ if and only if every $λ$ has absolute value less than $1$ . The matrix $A_{1}$ has eigenvalues of $λ_{1} = 1$ and $λ_{2} = - 0.3$ , while $A_{2}$ has eigenvalues of $λ_{1} = 0.3$ and $λ_{2} = 0.9$ both of which have absolute value less than 1.

So $A_{2}$ will has $A_{2}^{k} \to 0$ . While $A_{1}$ has an eigenvalue of $1$ , it won’t go to zero.

niuers

2020-03-20 00:00

Exercise 1.6.21

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