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.

Find the reciprocal condition number of the matrix.

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 $$Ax=b$$.

Now change $$b$$ by `0.01`

and solve the perturbed system.

Compare the solutions, `x`

and `x1`

.

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.