A Toeplitz matrix is a matrix whose entries are constant along each diagonal. The MATLAB command T = toeplitz(c,r) returns a Toeplitz matrix  having  as its first column and  as its first row. If the...

A Toeplitz matrix is a matrix whose entries are constant along each diagonal. The MATLAB command

T = toeplitz(c,r)

returns a Toeplitz matrix
 having
 as its first column and
 as its first row. If the first elements of
 and
 are different, a message is printed and the column element is used.

 Build the
pentadiagonal matrix

 Verify that the matrix is ill-conditioned.

Run the following experiment to observe the behavior of the eigenvalues of A as it is slightly perturbed. The MATLAB program computes the eigenvalues of a randomly perturbed A and outputs the norm of the difference between the eigenvalues of A and those of the perturbed matrix. The perturbations to the elements of A are in the range

_ij

⁻⁸.

 Are the eigenvalues of A ill-conditioned? For more information about this problem, see Ref. [29].