TABLE OF CONTENTS This appendix supplements Chapters 4 and 14 with concerns when the determinant of the coefficient matrix is small. We will introduce a reference against which the determinant can be measured to classify a matrix as a well- or ill-conditioned.
A norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. An example is the twodimensional Euclidean space denoted as R 2. The elements of the Euclidean vector space (e.g., (2,5)) are usually drawn as arrows in a two-dimensional cartesian coordinate system starting at the origin (0,0). The Euclidean norm assigns to each vector the length of its arrow.
The Euclidean norm of a matrix A, denoted as A, is defined as
and it is computed with the MATLAB function norm(A).
Using the MATLAB function norm(A), compute the Euclidean norm of the matrix A, defined as
Solution:
At the MATLAB command prompt, we enter
A=[2 5 ?4 9; ?3 ?6 8 1; 7 ?5 3 2; 4 ?9 ?8 ?1]; norm(A)
and MATLAB outputs
ans = 14.5539
The condition number of a matrix A is defined as
where A is the norm of the matrix A defined in relation (C.1) above. Matrices with condition number close to unity are said to be well- conditioned matrices
