STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

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and D λ has a non-negative diagonal. We define
a symmetric, p.s.d. square root of M by
M 12 = VD 12
λ
V0
— There are other roots, for instance P = VD 12
λ
W
for any orthogonal W (then PP 0 = M) butwe
will generally mean the one above.
• The rank of a real symmetric matrix equals the
number of non-zero eigenvalues. Reason: If
M = VD λ V 0 then the rank of M equals the
rank of D λ (why?), and the latter is clearly (is
it?) the number of non-zero diagonal elements.
— Note also that if M = VDV 0 is the spectral
decomposition then M and D have the same
eigenvalues, namely the diagonal elements of
D. This is because the characteristic polynomials
are the same:
¯
|M − I| = ¯V (D − I) V 0¯¯¯ = |D − I|