Matakuliah
Tahun
: Matrix Algebra for Statistics
: 2009
Patterned Matrix
Pertemuan 13
Patterned
Matrices that have a particular pattern
occur frequently in statistics. Such
matrices are typically used as
intermediary steps in proofs and in
perturbation techniques.
Patterned matrices also occur in
experimental designs and in certain
variance matrices of random vectors
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Some Identities
a) Identities that are useful, Assumed (that all
inverses exist)
i) VA-1(A - UD-1V) = (D - VA-1U)D-1V,
ii) D-1V(A - UD-lV)-1 = (D - VA-1U)VA-1
b) Setting A = I, D = -I, and interchanging U and V
in a) ii), we have that U(I + VU)-1= (I + UV)-LU
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Continued…
(c) If I + U is nonsingular,
(I + U)-1 = I - (I + U ) - W = I - U(I + U) -1
(d) U'A-1U(I + U'A-1U)-1= I - (I + U'A-1U)-1.
e) If A and B are n x n complex matrices,
then
In + AA' = (A + B)(In + B*B)-1(A + B)*
+ In- AB*)(I, +BB*)-1(In -AB*)*
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If A is nonsingular and the other matrices are
conformable square or rectangular matrices (e.g., A is
nxn, U is nxp , B is pxq, and V is qxn), then
(A + UBV)-1 = A-1 - (I + A-1UBV)-1A-1UBVA-1
=A-1 - A-1(I + UBVA-1)-1UBVA-1
=A-1 - A-1U(1 + BVA-1U)-1BVA-1
=A-1 - A-1UB(1 + VA-1UB)-1VA-1
=A-l - A-1UBV(1 + A-1UBV)-1A-1
=A-1 - A-1UBVA-1(I + UBVA-1)-1
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Gentle [1998: 621 notes that in linear regression
we often need inverses of various sums of
matrices and gives the following additional
identities for nonsingular A and B.
a) (A + BB')-lB = A-1B(I + B'A-1B) -1
b) (A-1 + B-l) -1 = A(A + B) -1B.
c) A(A + B) -1B = B(A + B)-lA.
d) A-1 + B-1 = A-l(A + B)B-1
We can also add, for nonsingular A + B,
e) A - A(A + B)-1A = B - B(A + B) -1B.
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