The Matrix Inverse
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:
Proof The Inverse Matrix The Inverse Matrix
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).
Example 1. Use
Gauss-Jordan elimination to find the inverse of 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 .
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.
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 .
Application to Continuous Least Squares
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
Gram matrix G is a matrix of inner products where the elements
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 .
Old Lab Project (Matrix Inversion Matrix Inversion). Internet hyperlinks to an old lab project.
Research Experience for Undergraduates
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.
Download this Mathematica Notebook The Matrix Inverse
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(c) John H. Mathews 2004