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Unformatted text preview: Y ∈ Sm , has the solution −
T
Y = X11 − X12 X221 X12 . (As usual, we take Sm as the domain of log det Y −1 .)
++
Hint. Use the Schur complement characterization of positive deﬁnite block matrices (page 651
of the book): if C ≻ 0 then
AB
0
BT C
if and only if A − BC −1 B T 0.
(c) Combine the result in part (b) and the minimization property (page 319, lecture notes) to
show that the function
−
T
f (X ) = log det(X11 − X12 X221 X12 )−1 , with dom f = Sn , is convex.
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−
T
(d) Show that (X11 − X12 X221 X12 )−1 is the leading m × m principal submatrix of X −1 , i.e.,
−
T
(X11 − X12 X221 X12 )−1 = P T X −1 P. Hence, the convex function f deﬁned in part (c) can also be expressed as f (X ) = log det(P T X −1 P ).
Hint. Use the formula for the inverse of a symmetric block matrix:
A
BT B
C −1 = 0
0
0 C −1 + −I
C −1 B T (A − BC −1 B T )−1 −I
C −1 B T T if C and A − BC −1 B T are invertible.
2.27 Functions of a random variable with logconcave densit...
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 Fall '13
 F.Borrelli
 The Aeneid

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