Patterned Matrix Pertemuan 13 Matakuliah : Matrix Algebra for Statistics Tahun : 2009
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
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
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*)*
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.