Elimination Methods

We motivated the need for elimination methods to compute determinants and to solve linear systems. Conceptually, the most important notion is that you view linear combinations to bring matrices into triangular form as matrix multiplications. This leads to the LU factorization of the matrix. Below is the output of a MATLAB session on the example we discussed in class.

The following material was produced via the diary command of MATLAB. 

% We illustrate the construction of the LU factorization
% of a 2-by-2 matrix with row reduction as you would do by hand
A = [1 2
     3 4]

A =

     1     2
     3     4

% we make the matrix A upper triangular,
% subtracting 3 times the first row from the second row:
U = A;
U(2,:) = U(2,:) - 3*U(1,:)

U =

     1     2
     0    -2

% this row reduction is represented by a multiplier matrix
M = eye(2);   % 2-by-2 identity matrix
M(2,1) = -3;
M*A

ans =

     1     2
     0    -2

% so the inverse of M is the lower triangular matrix L
% of the LU factorization, but no inverse operation is needed:
L = eye(2);
L(2,1) = 3;
L*U

ans =

     1     2
     3     4

% If we now pick a right hand side vector, say b = [0 2],
% then solving A*x = b amounts to solving two triangular systems
b = [0 2]'

b =

     0
     2

% first we solve L*y = b, followed by U*x = y:
y = L\b

y =

     0
     2

x = U\y

x =

     2
    -1

A*x

ans =

     0
     2

% Note that the built-in command for LU factorization
% produces a different answer than the one we obtained:
[l,u] = lu(A)

l =

    0.3333    1.0000
    1.0000         0


u =

    3.0000    4.0000
         0    0.6667

L  % the L we computed

L =

     1     0
     3     1

U

U =

     1     2
     0    -2

% What happened?  Well, MATLAB permuted the rows:
[l,u,p] = lu(a)

l =

    1.0000         0
    0.3333    1.0000


u =

    3.0000    4.0000
         0    0.6667


p =

     0     1
     1     0

% We will see in later lectures why pivoting is needed.