Let be an eigenvalue of a square matrix .
How to show that:
1. is an eigenvalue of where n is a positive integer.
2. If is invertible, show that is an eigenvalue of .
To start off: I was told to let be an eigenvector such that
....
Hello,
If is an eigenvalue of A, then there exists X such that :
Now multiply to the left by A :
Since is a scalar, it commutes with A.
Thus
Hence is an eigenvalue of
You can use this process to prove your question by induction.
2. If is invertible, show that is an eigenvalue of .
Multiply to the left by :
And same reasoning as above.
I did not phrase that question correctly.
What I meant:
if has eigenvalues
then will have eigenvalues
Based on the answer given above to the original question, it just seems that if it works for one eigenvalue, it will work for all.
When a matrix is raised to a power, will the eigenvalues also be raised to the same power?