MATRIX INVERSION

In order to find the inverse of a matrix, one can use Gauss Elimination by introducing as two right hand column vectors the columns of the unit matrix:

and obtaining the inverse matrix

Check this result:

It is not perfect, but good enough for the present purpose. What about the norms? They allow to talk about matrices as if they were simple numbers. There are different ways of doing this:

DEFINITIONS OF NORMS

A norm of a matrix satisfies the conditions:

It is known that these conditions are satisfied by the modulus or the length of a vector, also referred to as its Euclidean norm:

;

however, one could also employ instead the sum of the absolute values of its components x_i or even only their maximum values.

The norm of a matrix is defined in a corresponding manner in two steps. First form the sums of its columns or rows and then select their maximum:

Reconsider the last example and apply this approach to find its condition number. We had

,

whence

This explains the difficulties encountered when solving the equations.

PROBLEMS:

Solve by Gauss elimination at least one of the following systems of equations to 4 decimals:

Find the inverse of the matrix and the condition number. Do you consider the solution to be reliable?

L-U Decomposition

Another convenient method of solving systems of equations is by decomposition of the matrix into a product of upper and lower triangular matrices, and solving the system by combined forward and backwards substitutions. In L-U Decomposition of a 3x3 matrix, you have the product of two triangular matrices, each of which has six elements. However, the original matrix has only 9 elements. Hence one can set the diagonal elements of one of the triangular matrices equal to 1, for example, the diagonal elements of the upper matrix:

We see now that there are 9 equations expressing the a_ij in terms of the l_ij and u_ij. A special feature of this system is that one can find the solution explicitly and that, once an a_ij has been used, it can be overwritten. For this purpose, we need only follow the proper sequence:

You can now rewrite the original system:

and set

Hence, one has for the y_i the system:

which can be solved by forward substitution. Once the y_i have been found, the x_i are determined by backward substitution from the system defining the y_i.

EXAMPLE

Consider

We find:

Hence, the system for the auxiliary variables y_i is

with the solutions

and the x_i are the solutions of the system:

i.e.,

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