In May, I will be taking the qualifying exams for my Ph.D.. Over the next few weeks, I will be posting practice problems and my solutions to them. Until the end of February, I will be reviewing linear algebra, single variable real analysis, complex analysis and multivariable calculus. In March and April, I will be focusing on algebra, geometry and topology.
Here are three problems to start.
Problem: Suppose that is an real matrix with distinct real eigenvalues. Show that can be written in the form where each is a real number and the are real matrices with , and if . Give a real matrix for which such a decomposition is not possible and justify your answer.
Solution: for each , let denote the matrix with a on the entry and zeroes everywhere else. Then and when . Since has distinct real eigenvalues , it is diagonalizable over , so there is a real matrix such that , where . Let . Then
Moreover, for we have .
For the second part, notice that if the matrix is decomposed in the manner described above, the numbers are necessarily eigenvalues of . Indeed, multiplying the equality by and using that when , we find that . Hence, let be any nonzero vector. Since , at least one of the terms in the sum is nonzero, say . Then
and therefore is an eigenvector of with eigenvalue . Thus, it is impossible for the matrix to have such a decomposition if, say, it has no real eigenvalues, for example
Problem: Let be a continuous function on . Suppose there exists such that for all ,
Prove that .
Solution: (With Juan Ignacio Restrepo.) By iterated integration, we find by induction that, for every integer ,
Since the integrand is uniformly bounded and as , it follows that .
Problem: Suppose that is a real sequence with and that is a nonnegative sequence with . Prove that
Solution: For notational simplicity, let us write . Let be so large that for all . Then, for , we have, by the triangle inequality,
where . Let be so large that for all , which exists since . Then for , we have