I promised you some extra problems for Easter
1) Here is a simple one ... you might think:
Let x solve x' = x2. What happens when t grows?
(I have to use ' in place of the dot to denote time-derivative.)
2) A singular value of a matrix M, is a number s such that s2 is an eigenvalue of M'M (in this problem I use prime for transpose).
(a) Why does any matrix - square or not - have at least one real singular value?
(b) Even if we allow for eigenvalues to be complex, why are all singular values real?
(c) Can you think of a useful type of matrices for which we know that the singular values and the eigenvalues are precisely the same? If you don't spot it, do the next part first:
(d) We have, for square A, that (A2 - s2I) = (A - sI) (A + sI). Suppose that s is nonnegative and A does not have negative eigenvalues, when does then det(A - s2I) = 0?
3) Recall that exp(x) = ex can be written as 1 + x + x2 /2+ x 3/3! + ...
For square matrices A, define the matrix exponential as
exp(A) = I + A + A2/2 + A3/3! + ...
(a) Differentiate term by term (which is OK!): what is d/dt of exp(tA), where A does not depend on t? What is the derivative of exp(-tA)?
(b) Let x = x(t) (vector). Use the product rule to differentiate exp(-tA)x(t).
You should get exp(-tA)[x'(t) - Ax(t)] ... familiar?
(c) How would you go forth to solve the system x'(t) = Ax'(t) + b'(t)?
Sure that you catch all solutions this way ...?
We have issues here of course: actually calculating exp(tA) or exp(-tA) seems to be a formidable job. Except in some nicer cases:
(d) Suppose D is diagonal. Explain why exp(D) is diagonal, with exp(dii) as elements.
(e) Recall the text problems concerning matrices A which could be written AV = VD for D diagonal and V invertible. Suppose A admits this (it is called diagonalisation).
Show that exp(A) = V-1 exp(D) V.
(f) What is then exp(-tA)? And the derivative?
4) Old exam problems: Before even opening and glancing at old exam problem sets - there are way fewer in Mathematics 3 than Mathematics 2 - you should carefully consider that you might want to use a few to simulate an exam situation (highly recommended): Allocate three hours, open it at the beginning, do as much as you can. As of now there is no complete set you can do: you might be able to do 2/3 of the 2003 exam and of the 2007 exam; in both sets, you should skip 1b and 4, which we have not covered yet.