C = rcond(A) returns
an estimate for the reciprocal condition of A in
1-norm. If A is well conditioned, rcond(A) is
near 1.0. If A is badly conditioned, rcond(A) is
near 0.

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.

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