numerics – How to find the p-norm that meets the requirements?


On page 183 of this book there is Theorem 3:

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In other words, if the spectral radius of a matrix B is less than 1, there must be a norm $ ||B||_{p}$, so that $||B||_{p}<1$.

N(Eigenvalues(MatrixForm({{0, -1, 0}, {0, 1/2, 1}, {0, 1/5, 2/5}})))

But the following norms are all greater than 1:

N(Norm(MatrixForm({{0, -1, 0}, {0, 1/2, 1}, {0, 1/5, 2/5}}), 1))
N(Norm(MatrixForm({{0, -1, 0}, {0, 1/2, 1}, {0, 1/5, 2/5}}), 2))
N(Norm(MatrixForm({{0, -1, 0}, {0, 1/2, 1}, {0, 1/5, 2/5}}), Infinity))

What can I do to find the p-norm that makes $||A||_{p}<1$ hold?

Related exercises(2001武汉 岩石 数值分析):

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