If v is an eigenvector of A with eigenvalue , then . Taking of both sides, so that and . In other words, if v is an eigenvector of A with eigenvalue then it is also an eigenvector of with eigenvalue . Enough?
I have to find the Eigenvalues of a Matrix A. This is pretty easy but the second part of the question is:
What are the corresponding eigenvalues and eigenvectors for A^{3}-A^{-1}?
I know that the eigenvalues of A^{3} are equal to the third power of the eigenvalues of A, so this is no problem.
If the Eigenvalues of A are -2 and 3 like in my case, the Eigenvalues for A^{3} are -8 and 27.
But I don't know how to deal with the subtraction of the inverse, is there any rule or do I have to take the inverse of the matrix manually and subtract it from A^{3}?
In this case I would have to do a lot of matrix multiplication which takes a lot of time...
If v is an eigenvector of A with eigenvalue , then . Taking of both sides, so that and . In other words, if v is an eigenvector of A with eigenvalue then it is also an eigenvector of with eigenvalue . Enough?
well, let's see if that works:
suppose an eigenvalue of A is λ, with eigenvector v.
(A^{3} - I)(v) = A^{3}(v) - I(v) = A^{2}(Av) - v
= A^{2}(λv) - v = A(A(λv)) - v = A(λ(A(v))) - v = A(λ(λv)) - v = A(λ^{2}v) - v = λ^{2}(A(v)) - v
= λ^{2}(λv) - v = λ^{3}v - v = (λ^{3} - 1)v.
in other words, for all integers k, A^{k} has eigenvalue λ^{k} (even when k = 0) (if A^{k} is even defined for k < 0, that is, if A is invertible).
for example, let A be the matrix:
[2 4]
[0 2], which has eigenvalue 2. then A^{3} - I =
[7 48]
[0 ..7], which clearly has eigenvalue 7 = 2^{3} - 1.
[2 4]^{3}
[0 2] =
[2 4][2 4][2 4]
[0 2][0 2][0 2] =
[4 16][2 4]
[0 ..2][0 2] = (in the upper-right we have 2*4 + 4*2 = 8 + 8 = 16)
[8 48]
[0 ..8] (in the upper-right we have 4*4 + 16*2 = 16 + 32 = 48)
you don't just "cube each entry".
A^{3} *is* matrix multiplication (it's A times A times A). there's no simpler way to describe the entries of powers of a matrix, except by doing the multiplication (unless the matrix is diagonal, or of some other "special form").