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 Incoming links: Exponential of an idempotent linear map

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

Induction from

M^{2} = M

gives

M^{j} = M

for every integer

j \geq 1

. Separate the zeroth term of the matrix exponential series:

e^{α M} = I + j = 1 \sum \infty \frac{α ^{j} M ^{j}}{j !} = I + M j = 1 \sum \infty \frac{α ^{j}}{j !} = I + (e^{α} - 1) M .

Therefore the exponential of an idempotent linear map is

e^{α M} = I + (e^{α} - 1) M = (I - M) + e^{α} M .

The last form shows its action directly: it multiplies the image of a linear map of

M

by

e^{α}

and fixes its kernel of a linear map. No Hermiticity assumption on the idempotent matrix is needed.

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