Answer :
You can use the definition of an eigenvalue and eigenvector to prove that Ak has eigenvalue λk for the same eigenvector x for any positive integer k.
You can use the eigenvalue definition Ax = λx to prove that Ak has eigenvalue λk for the same eigenvector x for any positive integer k.
To begin, let's start with the definition of an eigenvalue and eigenvector:
x = λx
where A is a matrix, λ is the eigenvalue, and x is the eigenvector.
Now, let's raise both sides of this equation to the power of k:
[tex](Ax)^k = (\lambda x)^k[/tex]
Since [tex](A^k)x = A(Ax)^{(k-1)[/tex], we can substitute and simplify:
[tex]A^{kx} = \lambda(Ax)^{(k-1)[/tex]
Substituting the definition of an eigenvalue and eigenvector again, we get:
[tex]A^{kx} = \lambda^{kx[/tex]
This shows that Ak has eigenvalue λk for the same eigenvector x, as desired.
Therefore, you can use the definition of an eigenvalue and eigenvector to prove that Ak has eigenvalue λk for the same eigenvector x for any positive integer k.
To learn more about eigenvector, visit:
brainly.com/question/29768825
#SPJ4
