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.

norm(x-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.

det(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.

rcond(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.