Since is a diagonal matrix with diagonal entries , its inverse, denoted as , is also a diagonal matrix. The diagonal entries of are the reciprocals of the diagonal entries of :
To find the inverse matrix of , use the given equation . Multiply both sides by and appropriately:
Taking the inverse of both sides, we get:
Using the property of inverses for matrix products:
Given the equation , this implies that is the diagonal form of under the similarity transformation by . The diagonal entries of , denoted as , are the eigenvalues of .
To show this formally, consider , where is the -th standard basis vector. We have:
Since , let . Then:
Hence, is an eigenvector of corresponding to the eigenvalue .
From the similarity transformation , raising both sides to the power gives:
Since :
Because is diagonal, is also diagonal, with each diagonal element being raised to the power :
Thus, has the same eigenvectors as , and the eigenvalues are the -th powers of the eigenvalues of . Therefore, the eigenvalue-eigenvector pairs for are:
Eigenvalue:
Corresponding eigenvector:
In summary, every eigenvalue of raised to the power is an eigenvalue of , and the corresponding eigenvectors remain the same.