Page 103 - Applied Numerical Methods Using MATLAB

92 SYSTEM OF LINEAR EQUATIONS
(2) (2)
second row r and r to get
1 2
(2) (3)  (3) (3) (3) (3)    (3)
r − 0 × r → a a a b 1 001 = x 1 : r
1 3 11 12 13 1 1
(2) (3)  (3) (3) (3)    (3)
r − (−1) × r →  a a a b (3)  =   : r
2 3  21 22 23 2   0 101 = x 2  2
(2) (3) (3) (3) (3) (3)
r → a a a b : r
3 31 32 33 3 0 011 = x 3 3
(2.2.30c)
After having the identity matrix–vector form like this, we take the RHS vector
as the solution.
The general formula applicable for Gauss–Jordan elimination is the same as
Eq. (2.2.5), except that the index set is m = k—that is, all the numbers from
m = 1to m = M except m = k. Interested readers are recommended to make
their own routines to implement this algorithm (see Problem 2.3).

2.3 INVERSE MATRIX
In the previous section, we looked over some algorithms to solve a system of
linear equations. We can use such algorithms to solve several systems of linear
equations having the same coefficient matrix

Ax 1 = b 1 ,Ax 2 = b 2 ,...,Ax NB = b NB

by putting different RHS vectors into one RHS matrix as

b 2 ··· b NB ], AX = B
A[ x 1 x 2 ··· x NB ] = [ b 1
(2.3.1)
−1
X = A B
If we substitute an identity matrix I for B into this equation, we will get the matrix
−1
−1
inverse X = A I = A . We, however, usually use the MATLAB command
inv(A) or A^-1 to compute the inverse of a matrix A.

2.4 DECOMPOSITION (FACTORIZATION)

2.4.1 LU Decomposition (Factorization): Triangularization
LU decomposition (factorization) of a nonsingular (square) matrix A means
expressing the matrix as the multiplication of a lower triangular matrix L and
an upper triangular matrix U, where a lower/upper triangular matrix is a matrix
having no nonzero elements above/below the diagonal. For the case where some
row switching operation is needed like in the Gauss elimination, we include a
permutation matrix P representing the necessary row switching operation(s) to
write the LU decomposition as
PA = LU (2.4.1)