Show that if A is an n x n matrix, then A" can be written as a linear combination of the matrices I, A, A², . . , A" = a0I + a1A+¤2A² + An-1; that is, there are scalars ao, ɑ1, a2, . .. , an-1 such that + an-14"-1.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Hint: Cayley-Hamilton Theorem

Show that if A is an n x n matrix, then A" can be written as a linear combination of
the matrices I, A, A², ... , A"-1; that is, there are scalars ao, a1, a2, ..., an-1 such that
A" = aoI + a1A+¤2A² + •
+ an-1A"-1.
•..
Transcribed Image Text:Show that if A is an n x n matrix, then A" can be written as a linear combination of the matrices I, A, A², ... , A"-1; that is, there are scalars ao, a1, a2, ..., an-1 such that A" = aoI + a1A+¤2A² + • + an-1A"-1. •..
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