Monday, 31 March 2008

Schur's Lemma (2)

Here's the follow up post to Schur's Lemma, in case you were anxiously awaiting it.

The corollary says that if α=β, f has to be a multiple of the identity function (or unit matrix).

For the proof one looks at an eigenvalue λi. And subtracts the following on both sides:

This changes to:

Now we apply the Schur's Lemma from last time and find out that has to be either invertible or the 0 function. But it cannot be invertible because λi is an eigenvalue (actually the only eigenvalue). And we have what we wanted.

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