ill-conditioned and well-conditioned matrices?

Thurman Buege: The condition of a matrix iskappa(A) = ||A|| ||inv(A)||and technically, the determinant is unrelated.Any orthogonal matrix has a determinant of 1 (all it's eigenvalue are on the unit circle) and its condition number is also 1 which is as well conditioned as possible. You can make a simple orthogonal 2x2 matrix with a rotation:Choose an angle theta in [-pi,pi]. The matrix A = [cos(theta), sin(theta); -sin(theta), cos(theta)] is orthogonal.Another easy matrix whose determinant is small but is well conditioned is a multiple of the identity. It's condition is also 1, but if the terms along the diagonal are less than 1 the determinant of the matrix approaches zero. Conversely, if the multiple you use is greater than one, the determinant of the matrix tends to infinity but the matrix is still well conditioned.Most large, dense matrices are ill-conditioned and have large determinants. One specifically crappy group is the vandermonde matrices (check out the so! urce for how to make them).If you wanted a matrix with a small determinant but large condition number, there is a common test case (I don't know if it has a name) which is the unsymmetric matrix:[-149, -50, -154; 537, 180, 546; -27, -9, -25]Good luck. Glad to see people doing matrix computations....Show more

Gilberto Cratin: Yes. If a matrix is ill-conditioned, then a small change in the argument can cause a large change in its image. That makes it an ill-posed problem.