Let A be an invertible nxn matrix and v an eigenvector of A with associated eigenvalue λ.?

a) Is v an eigenvector of A^3? if so, what is the eigenvalue?

b) Is v an eigenvector of A^-1? if so, what is the eigenvalue?

c) Is v an eigenvector of A + 2I? if so, what is the eigenvalue?

d) is v an eigenvector of 7A? if so, what is the eigenvalue?

e) Let A be an nxn matrix and let B=A-αI for some scalar α. How do the eigenvalues of A and B compare?

Make sure to Justify your answers!! Thanks!

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  • 7 years ago
    Best Answer

    for v to be an eigenvector with eigenvalue λ

    A v = λ v

    a) A v = λ v

    multiply by A

    A^2 v = A λ v = λ A v = λ λ v = λ^2 v

    A^3 v = A λ^2 v =λ^2 A v = λ^3 v

    so v is an eigenvector of A^3 and λ^3 is the associated eigenvalue

    b) A v = λ v

    left multiply by A^-2

    A^-2 A v = A^-2 λ v

    A^-1 v = λ A^-2 v = (λ A^-2) v

    for v to be an eigenvector of A^-1 then A^-2 must = I the unit matrix

    so v is not in general an eigenvector of A^-1

    c) A v = λ v

    add 2 I v to both sides

    Av + 2 I v = λ v + 2 I v or noting that 2 I v = 2 v on the right hand side

    (A+2 I ) v = (λ + 2 ) v

    so v is an eigenvector of Av + 2 I v with associated eigenvalue of λ + 2

    d) A v = λ v

    7 A v = 7 λ v

    (7A) v = (7 λ) v

    so v is an eigenvector of 7A and has eigenvalue of 7 λ

    e) A v = λ v and

    B=A-αI

    B v = A v - α v

    B v = λ v - α v = (λ - α) v

    so the eigenvalue of B called λb is

    λb = λ - α

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