Question

What happens to the eigenvalues and eigenvectors of a specific matrix when that matrix is put to the power of a number (e.g. 5)?

Answer 1

The eigenvalues of a matrix raised to a power are themselves raised to that power, while the eigenvectors remain unchanged. For instance, if λ is an eigenvalue of matrix A, then λ to the fifth power will be an eigenvalue of A to the fifth power, with the corresponding eigenvectors staying consistent.

Step-by-step explanation:

The eigenvalues of a matrix when raised to a power, such as the fifth power, are raised to that same power. If λ is an eigenvalue of matrix A, then λ5 will be an eigenvalue of the matrix A5. As for the eigenvectors, they remain the same; if v is an eigenvector of A corresponding to λ, then v will also be an eigenvector of A5 corresponding to λ5. This concept can be generalized to any integer power of the matrix.