 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015/4/solution

B (u) B (v) = B (u + v), B (u) = (cosh u sinh u sinh u cosh u) .

Since

M = B (y) /2

, its fourth matrix power is

M^{4} = \frac{1}{16} (cosh 4 y sinh 4 y sinh 4 y cosh 4 y) .

Alternatively, the orthonormal eigenvectors

(1, \pm 1) / 2

e^{\pm y} /2

. Raising these eigenvalues to the fourth power gives the same result by spectral decomposition.

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