MATH307 April 2005
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Find for any positive integer when is the matrix
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Recall that if is diagonalisable, then . Thus, we seek the eigenvalues of to form the diagonal matrix , and the eigenvectors of to make and .
To find the eigenvalues, recall that Hence we find the characteristic polynomial and set to 0: This implies that we have two eigenvalues as required:
Now we must find the corresponding eigenvectors
Now we can construct and find where the columns of are the eigenvectors and the diagonal entries are the corresponding eigenvalues:
It follows that