# 950 Extra Problems Modules

## Modules and Linear Algebra

• Prove the Cayley-Hamilton theorem.
• Prove that the minimal polynomial divides the characteristic polynomial.
• Prove that the cokernel of $$A\in \operatorname{Mat}(n\times n, {\mathbf{Z}})$$ is finite $$\iff \operatorname{det}A \neq 0$$, and show that in this case $${\left\lvert {\operatorname{coker}(A)} \right\rvert} = {\left\lvert {\operatorname{det}(A)} \right\rvert}$$.
• Show that a nilpotent operator is diagonalizable.
• Show that if $$A,B$$ are diagonalizable and $$[A, B] = 0$$ then $$A,B$$ are simultaneously diagonalizable.
• Does diagonalizable imply invertible? The converse?
• Does diagonalizable imply distinct eigenvalues?
• Show that if a matrix is diagonalizable, its minimal polynomial is squarefree.
• Show that a matrix representing a linear map $$T:V\to V$$ is diagonalizable iff $$V$$ is a direct sum of eigenspaces $$V = \bigoplus_i \ker(T -\lambda_i I)$$.
• Show that if $$\left\{{\mathbf{v}_i}\right\}$$ is a basis for $$V$$ where $$\dim(V) = n$$ and $$T(\mathbf{v}_i) = \mathbf{v}_{i+1 \operatorname{mod}n}$$ then $$T$$ is diagonalizable with minimal polynomial $$x^n-1$$.
• Show that if the minimal polynomial of a linear map $$T$$ is irreducible, then every $$T{\hbox{-}}$$invariant subspace has a $$T{\hbox{-}}$$invariant complement.