Reciprocal condition number
Examine the sensitivity of a badly conditioned matrix.
A notable matrix that is symmetric and positive definite, but badly conditioned, is the Hilbert matrix. The elements of the Hilbert matrix are H(i,j) = 1/(i + j -1).
Create a 10-by-10 Hilbert matrix.
A = hilb(10);
Find the reciprocal condition number of the matrix.
C = rcond(A)
C = 2.8286e-14
The reciprocal condition number is small, so A is badly conditioned.
The condition of A has an effect on the solutions of similar linear systems of equations. To see this, compare the solution of Ax = b to that of the perturbed system, Ax = b + 0.01.
Create a column vector of ones and solve Ax = b.
b = ones(10,1); x = A\b;
Now change b by 0.01 and solve the perturbed system.
b1 = b + 0.01; x1 = A\b1;
Compare the solutions, x and x1.
ans = 1.1250e+05
Since A is badly conditioned, a small change in b produces a very large change (on the order of 1e5) in the solution to x = A\b. The system is sensitive to perturbations.
Examine why the reciprocal condition number is a more accurate measure of singularity than the determinant.
Create a 5-by-5 multiple of the identity matrix.
A = eye(5)*0.01;
This matrix is full rank and has five equal singular values, which you can confirm by calculating svd(A).
Calculate the determinant of A.
ans = 1.0000e-10
Although the determinant of the matrix is close to zero, A is actually very well conditioned and not close to being singular.
Calculate the reciprocal condition number of A.
ans = 1
The matrix has a reciprocal condition number of 1 and is, therefore, very well conditioned. Use rcond(A) or cond(A) rather than det(A) to confirm singularity of a matrix.
Reciprocal condition number, returned as a scalar. The data type of C is the same as A.
The reciprocal condition number is a scale-invariant measure of how close a given matrix is to the set of singular matrices.
If C is near 0, the matrix is nearly singular and badly conditioned.
If C is near 1.0, the matrix is well conditioned.