Problem 17

The matlab command hilb(n) computes the Hilbert matrix of size \(n \times n\). What are the eigenvalues and eigenvectors of this matrix, for various values of \(n\)? What can you say about the condition number?

Solution

Matlab

Computing the eigenvalues and eigenvectors of a matrix is performed via eig:

[V,E] = eig( hilb(3) );

Show the eigenvalues:

diag(E)
ans =

  0.0026873
  0.1223271
  1.4083189

And the eigenvectors:

V
V =

  -0.12766   0.54745   0.82704
  0.71375  -0.52829   0.45986
  -0.68867  -0.64901   0.32330

The condition number of a matrix is computed with the command cond:

for i=1:10
  printf( "%d : %.6e\n", i, cond(hilb(i)) )
end

Output:

1 : 1.000000e+00
2 : 1.928147e+01
3 : 5.240568e+02
4 : 1.551374e+04
5 : 4.766073e+05
6 : 1.495106e+07
7 : 4.753674e+08
8 : 1.525758e+10
9 : 4.931538e+11
10 : 1.602442e+13

Mathematica

There is a built-in command called HilbertMatrix:

HilbertMatrix[3] // MatrixForm

Compute the eigenvalues and eigenvectors of the matrix (output is numerical):

Eigensystem[HilbertMatrix[3]] // N
Eigensystem[HilbertMatrix[4]] // N

How long does it take for \(n=10\):

Timing[Eigensystem[HilbertMatrix[10]]][[1]]

A table of condition numbers, \(n\) goes from 1 to 10:

Table[LinearAlgebra`MatrixConditionNumber[HilbertMatrix[n]], {n, 1, 10}]