Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 40E / a

Solution

The

m

K_{m} (A, v) = span {v, A v, \dots, A^{m - 1} v} .

Because

A

has a complete eigenbasis, decompose

v = v_{1} + \dots + v_{s},

where

v_{j}

is the projection of

v

onto the eigenspace belonging to the

j

th distinct eigenvalue

λ_{j}

; some

v_{j}

may vanish. Then

A^{k} v = \sum_{j = 1}^{s} λ_{j}^{k} v_{j} .

Every generator of

K_{m} (A, v)

therefore belongs to

span {v_{1}, \dots, v_{s}}

, and

dim K_{m} (A, v) \leq s for every m .

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