1. (a) Suppose that A is an orthogonal matrix. What are its singular values?
(b) Is the SVD of a given matrix A unique in general?
2. Obtain the matrix A for Example 4.17 in the case where v'(0) = v'(1) = 0, and show that it is singular.
Example 4.17
(differential equations). Consider the recovery of a function v(t) from its given second derivative -g(t) on the interval [0,1]. Such a problem is typical for many applications. An example is the celebrated heat equation in one dimension: if the intensity of a heat source is given and we wish to recover the steady-state temperature of the body that is being heated, then under certain simplifying assumptions about the heat conductivity properties of the material we obtain a problem of the same type.