Homework 6

(Additional problems)

Problem 1   A matrix $A\in M_{n\times n}(F)$ is called nilpotent if $A^{n}=0$ for some $n\neq 0$. Show that if a matrix is nilpotent, then the only eigenvalue of $A$ is $0$.

Problem 2   Prove that the eigenvalues of an upper-triangular matrix are its diagonal entries.

Problem 3   Prove that a linear operator on an $n$-dimensional vector space which has $n$ distinct eigenvalues is diagonalizable.

Problem 4   Prove that if $(I-AB)$ for $A,B\in M_{n\times n}(F)$ is invertible, then so is $(I-BA)$, and $(I-BA)^{-1}=I+B(I-AB)^{-1}A$. (You might want to use this result in problem 6 on page 106).

Problem 5   Let $V=C(\mathbb{R},\mathbb{R})$ be the vector space of all continuous real-valued functions on the real line. Prove that the operator $T:V\to V$ given by $(Tf)(x)=\int _{0}^{x}f(t)dt$ has no eigenvalues.

Problem 6   a) Give a matrix (with real entries) which is diagonalizable over $\mathbb{C}$, but is not diagonalizable over $\mathbb{R}$.

b) Give an example of a matrix which is not diagonalizable over $\mathbb{C}$.
(Hint: in both parts, it can be useful to think first about an operator with such properties).