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Patterned Matrix Pertemuan 13

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Patterned Matrix Pertemuan 13

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Patterned Matrix Pertemuan 13

Matakuliah: Matrix Algebra for Statistics

Tahun: 2009

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

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

(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*)*

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

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.