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Matrix proof

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Prove that

||x^(k) - x|| <= (||T||^k)(||x^(0) - x||) and
||x^(k) - x|| <= (||T||^k/(1-||T||))(||x^(1)-x^(0)||),

where T is an n x n matrix with ||T|| < 1 and

x^(k)=Tx^(k-1)+c, k=1,2,...,

with x^(0) arbitrary, c belonging to R^n, and x=Tx+c.

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Solution Summary

The solution provides a proof regarding matrices.

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