Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 7B / b

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- Solution b

Solution

The characteristic polynomial of

A

is

χ_{A} (t) = det (t I - A) .

λ \in C

is an eigenvalue when there is a nonzero vector

v

such that

A v = λ v .

Such a

v

is an eigenvector belonging to

λ

. Equivalently,

λ

is a root of

χ_{A}

and the corresponding eigenvectors are the nonzero elements of

ker (A - λ I)

.

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