Question: If A1, A2, , Ak, is an infinite sequence of n Ã n matrices, then the sequence is said to converge to the n Ã

If A1, A2, ˆ™ ˆ™ ˆ™, Ak, ˆ™ ˆ™ ˆ™ is an infinite sequence of n Ã— n matrices, then the sequence is said to converge to the n Ã— n matrix , A if the entries in the ith row and jth column of the sequence converge to the entry in the ith row and jth column of A for all i and j. In that case we call A the limit of the sequence and write limk†’ + ˆž Ak = A. The algebraic properties of such limits mirror those of numerical limits. Thus, for example, if P is an invertible n Ã— n matrix whose entries do not depend on k, then

if and only if

(a) Suppose that A is an n Ã— n diagonalizable matrix. Under what conditions on the eigenvalues of A will the sequence A, A2, . . ., Ak, . . . converge? Explain your reasoning.
(b) What is the limit when your conditions are satisfied?

lira Ak-A im PAPP AP

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a Since A is diagonalizable there exists an invertible matrix P such that P1 AP D where D is a diago... View full answer

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