For a system of equations expressed as
A*x = b;
generally, you should use x = inv(A)*b only if the system is of full rank and is well-conditioned. The inverse of A exists only for well defined full-rank matrices and thus cannot be used for over- or under-defined systems. You should use x=A\b for an over- or under-defined system, or one which is badly conditioned.
For a well defined system, both the methods will yield the same result. The following example illustrates the above:
rand('state',0)
A = round(10*rand(4,4));
b = round(20*rand(4,1));
The A and b define a system of equations A*x = b. The RANK function
rank(A)
ans =
4
shows that A is a well defined system. The COND function
cond(A)
ans =
11.1361
shows that A is fairly well-conditioned. Thus to obtain the solution of the system A*x = b, both the INV function and the backslash operator can be used.
x1 = inv(A)*b
x1 =
-1.4378
-0.0922
2.2467
1.8038
x2 = A\b
x2 =
-1.4378
-0.0922
2.2467
1.8038
Even though both of the above procedures yield similar results for a well-defined and well-conditioned system, we recommend that you use the '\' operator to solve a system of linear equations, as it will handle both well-conditioned and ill-conditioned matrices robustly and can handle over- or under-determined systems. INV requires the coefficient matrix to be square.
rand('state', 0)
A = [round(10*rand(3,4))]; % an over determined system
b = round(10*rand(3,1));
For this example, you cannot use x = inv(A)*b because A is not square. In this case you will have to use x = A\b.
x = A\b
x =
1.2400
0.5022
-1.1822
0
More information on the \ operator can be found by typing HELP SLASH at the MATLAB command prompt. More information on kinds of systems is given below.
A system of linear equations can be expressed as one of the following:
(a) Well-defined systems: The system where the number of unknown variables are the same as the number of linearly independent equations (LIE) in the system.
(b) Under-defined systems: The system where the number of unknown variables exceeds the number of LIE relating these variables.
(c) Over-defined systems The system where the number of unknown variables are less then the number of LIE relating these variables.
Mathematically, if A*x = b, then the definition of the system can be obtained as follows:
If A is a square matrix and the rank of A is equal to the length of the matrix, then the matrix is well defined. If A is of size m x n and m>n, then the system is underdefined. If m<n, the system is overdefined.
