STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

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— The trace of a square matrix is the sum of
its diagonal elements. You should verify that
(AB) = (BA). Thus products within
traces can be rearranged cyclically:
(ABC) =(CAB) = (BCA)
but not necessarily = (ACB)
It will be shown that for an idempotent matrix,
= . A consequence is that
(I − H) = (I − H) = − (H)
µ
= − X ³ X 0 X´−1
X
0
= −
µX 0 X ³ X 0 X´−1
= −
Similarly (X) =(H), (H) =.
• Gram-Schmidt Theorem: Every -dimensional
vector space has an orthonormal basis.
Proof: Start with any basis v 1 v . Normalize
v 1 to get a unit vector (i.e. a vector with unit
norm) q 1 ∈ ; in general suppose that mutually