Page 480 - Applied Numerical Methods Using MATLAB

MATRIX INVERSION LEMMA 469
Note also that the principal minor matrices are the submatrices taking the diagonal
elements from the diagonal of the matrix A and, say for a 3 × 3matrix, the
principal minor matrices are

 

a 11 a 12 a 13
a 11 a 12 a 22 a 23 a 11 a 13
a 11 ,a 22 ,a 33 , , , ,  a 21 a 22 a 23 
a 21 a 22 a 32 a 33 a 31 a 33
a 31 a 32 a 33
among which the leading ones are
 
a 11 a 12 a 13

a 11 a 12
a 11 , ,  a 21 a 22 a 23 
a 21 a 22
a 31 a 32 a 33
B.13 SCALAR (DOT) PRODUCT AND VECTOR (CROSS) PRODUCT
A scalar product of two N-dimensional vectors x and y is denoted by x · y and
is defined by
N
T
x · y = x n y n = x y (B.13.1)
n=1
T
An outer product of two three-dimensional column vectors x = [x 1 x 2 x 3 ] and
T
y = [y 1 y 2 y 3 ] is denoted by x × y and is defined by
 
x 2 y 3 − x 3 y 2
x × y =  x 3 y 1 − x 1 y 3  (B.13.2)
x 1 y 2 − x 2 y 1

B.14 MATRIX INVERSION LEMMA

−1
Matrix Inversion Lemma. Let A, C,and [C −1 + DA B] be well-defined with
nonsingularity as well as compatible dimensions. Then we have
−1
−1
−1
[A + BCD] −1 = A −1 − A B[C −1 + DA B] DA −1 (B.14.1)
Proof. We will show that postmultiplying Eq. (B.14.1) by [A + BCD] yields an
identity matrix.

−1
−1
−1
−1
[A −1 − A B[C −1 + DA B] DA ][A + BCD]
−1
−1
−1
−1
= I + A BCD − A B[C −1 + DA B] D
−1
−1
−1
−1
− A B[C −1 + DA B] DA BCD