Module

for

The Matrix Inverse

Background

Theorem (
Inverse Matrix) Assume that is an nonsingular matrix. Form the augmented matrix of dimension  .  Use Gauss-Jordan elimination to reduce the matrix so that the identity is in the first columns.  Then the inverse is located in columns .  The augmented matrix looks like:

We can use the previously developed Gauss-Jordan subroutine to find the inverse of a matrix.

Computer Programs  The Inverse Matrix  The Inverse Matrix

Algorithm  (Complete Gauss-Jordan Elimination).  Construct the solution to the linear system    by using Gauss-Jordan elimination.  Provision is made for row interchanges if they are needed.

Mathematica Subroutine (Complete Gauss-Jordan Elimination).

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Example 1.  Use Gauss-Jordan elimination to find the inverse of the matrix  .
Solution 1.

Definition (Hilbert Matrix).  The elements of the Hilbert matrix   of order n are    for    and  .

Some Hilbert Matrices.

Example 2.  Find the inverse of the 5×5 Hilbert matrix  .
Solution 2.

Example 3.  Hilbert matrices are known to be ill-conditioned. Consider the matrix A given by

Find the inverse of the 5×5 matrix A that approximates the inverse of the  5×5 Hilbert matrix .
Remark. The entries in the matrix for this exercise must be typed in by hand in order to make sure that only six decimal places are stored in the computer.
Solution 3.

The Inverse Hilbert Matrix

The formula for the elements of the inverse Hilbert matrix   of order  n  is known to be

which can be expressed using binomial coefficients

.

When exact computations are needed these formulas should be used instead of using a subroutine or built in procedure for computing the inverse of  .
Verification.

Application to Continuous Least Squares Approximation

The continuous least squares approximation to a function   on the interval [0,1] for the set of functions    can solved by using the normal equations

(1)           for   .

Where the inner product is .  Solve the linear system (1) for the coefficients and construct the approximation function

.

Definition (Gram Matrix).  The Gram matrix G is a matrix of inner products where the elements are  .

The case when the set of functions is    will produce the Hilbert matrix.  Since we require the computation to be as exact as possible and an exact formula is known for the inverse of the Hilbert matrix, this is an example where an inverse matrix comes in handy.

Example 4.  Find the continuous least squares polynomial of degree n=4 that approximates the function    over the interval  .
Solution 4.

Old Lab Project (Matrix Inversion  Matrix Inversion).   Internet hyperlinks to an old lab project.

The Inverse Matrix  The Inverse Matrix  Internet hyperlinks to web sites and a bibliography of articles.

The Hilbert Matrix  The Hilbert Matrix   Internet hyperlinks to web sites and a bibliography of articles.

(c) John H. Mathews 2004