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.

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.

This matrix is full rank and has five equal singular values,
which you can confirm by calculating `svd(A)`

.

Calculate the determinant of `A`

.

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`

.

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.