Two 3D-problems

In this exercise you will practice drawing and interpreting three-dimensional objects.

Question A square matrix, A, of order n has n2 elements and we can regard it as a point  in a space of dimension n2. The set of all singular matrices (of order n) form a surface in the space and a particular singular matrix is a point on the surface. If we wish to visualize this surface, there will problems even for small n. If we assume that n = 2 and that the matrix is real and symmetric there are only three free parameters (elements) in the matrix, and we have a chance of drawing the surface. Visualize the surface in this case. We would like to see the surface in a neighbourhood of the origin (so in all eight octants). Mark also the matrices [1 0; 0 1] and [1 0; 0 0.01] and the corresponding closest (in || ||2) singular matrices, which are [1 0;0 0] in both cases (though not unique in the first).
I have sometimes used this picture, when talking about the condition number, k(A) = || A || || A-1 ||, for the Ax=b-problem. One can show that 1 / k(A) is the distance from A to the closest singular matrix (at least for some norms). So an ill-conditioned problem, large k(A), has a matrix which is close to the surface. A nice, well-conditioned problem, small k(A), has a matrix that lies far from the surface.

To think about (you do not have to do this in Matlab): can we visualize "surface, distance, closest", when n = 2 but when the matrix is unsymmetric?


Question You have a polynomial, x3 + a x2 + b x + c, where a, b and c are real numbers. Regard (a, b, c) as a point in 3D and visualize the set of points where the polynomial has multiple zeros (of multiplicity two or three). From your plot we should, in particular, be able to see the points where the polynomial has a zero of multiplicity three (i.e. three identical zeros). It is important to think a little bit first in this problem, it is easy if you do it the right way. The resulting graphics should look smooth.

For what set of (a, b, c) are all the roots real? For what set do we have at least one zero whose imaginary part is non-zero? You do not have to plot anything to answer the two last questions, just use your plot and think. It may be useful to know that the zeros of a polynomial are continuous functions of the coefficients of the polynomial.
 
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