 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015-resit/11/b/solution

A power of a diagonal matrix acts entrywise. Thus

D^{n} = diag ((- 1)^{n}, 2^{n}, 1) .

From

A = B D B^{- 1}

, adjacent

B^{- 1} B

factors cancel in repeated multiplication, giving

A^{n} = B D^{n} B^{- 1} .

This is the matrix power rule for a similarity transformation, proved directly rather than by taking powers of separate noncommuting factors.

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