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Past exam of the mathematical methods I course of the University of Oxford / 2019 / 11 / b / Solution 2026-10-25

The upper-triangular matrix has distinct eigenvalues

2, - 1

, with respective eigenvectors

(1, 0)^{T}

and

(- 2, 1)^{T}

. Thus

A = S (20 0 - 1) S^{- 1}, S = (10 - 2 1), S^{- 1} = (1021) .

The defining power series shows that the matrix exponential respects similarity. Applying solution to the diagonal factor gives

e^{A} = S (e^{2} 0 0 e^{- 1}) S^{- 1} = (e^{2} 0 2 (e^{2} - e^{- 1}) e^{- 1}) .

As a direct series check,

(A^{m})_{12} = 2 (2^{m} - (- 1)^{m})

for

m \geq 0

; summing this entry divided by

m!

gives the same off-diagonal expression.

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