Why do I get the following warning message:
Warning: Matrix is close to singular or badly scaled.
MATLAB: Do I get warning message ‘Matrix is close to singular or badly scaled. ‘
MATLAB
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DON'T use det to determine if a matrix is singular!
A non-zero multiple of the identity matrix isn't singular, right?
A = 0.1*eye(400);det(A)The determinant of A underflowed to 0.
Can multiplying a singular (or nearly singular) matrix by a non-zero scalar make it no longer singular?
B = [1 1; 1 1+eps];C = 1e8*B;det(B)det(C)A number with a determinant on the order of 1 can't be singular, can it?
Use the cond or rcond functions instead. Matrices with condition numbers closer to 1 are better behaved.
cond(A)cond(B)cond(C)If you're calling inv to try to solve a system of linear equations, DON'T. Use the backslash operator (\) instead. Even if your textbook told you to multiply by the inverse matrix, well, in theory there is no difference between theory and practice ...
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 = 4shows that A is a well defined system. The COND function
cond(A) ans = 11.1361shows 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 1.8038x2 = A\b x2 = -1.4378 -0.0922 2.2467 1.8038Even 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 0More 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.
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