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, soAis badly conditioned.

The condition ofAhas 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, $$Ax=b+0。01$$。

创建一个列向量的和求解e $$Ax=b$$。

b = ones(10,1); x = A\b;

Now change $$b$$by0.01and solve the perturbed system.

b1 = b + 0.01; x1 = A\b1;

Compare the solutions,xandx1。

norm(x-x1)

ans = 1.1250e+05

SinceAis badly conditioned, a small change inbproduces a very large change (on the order of 1e5) in the solution tox = A\b。系统m 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 calculatingsvd(A)。

Calculate the determinant ofA。

det(A)

ans = 1.0000e-10

Although the determinant of the matrix is close to zero,Ais actually very well conditioned andnotclose to being singular.

Calculate the reciprocal condition number ofA。

rcond(A)

ans = 1

The matrix has a reciprocal condition number of1and is, therefore, very well conditioned. Usercond(A)orcond(A)rather thandet(A)to confirm singularity of a matrix.