| course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
39 |
|
Paper 1, Section II, F |
2012 |
Define what it means for two $n \times n$ matrices to be similar to each other. Show that if two $n \times n$ matrices are similar, then the linear transformations they define have isomorphic kernels and images.
If $A$ and $B$ are $n \times n$ real matrices, we define $[A, B]=A B-B A$. Let
$$\begin{aligned}
K_{A} &=\left{X \in M_{n \times n}(\mathbb{R}) \mid[A, X]=0\right} \\
L_{A} &=\left{[A, X] \mid X \in M_{n \times n}(\mathbb{R})\right}
\end{aligned}$$
Show that $K_{A}$ and $L_{A}$ are linear subspaces of $M_{n \times n}(\mathbb{R})$. If $A$ and $B$ are similar, show that $K_{A} \cong K_{B}$ and $L_{A} \cong L_{B}$.
Suppose that $A$ is diagonalizable and has characteristic polynomial
$$\left(x-\lambda_{1}\right)^{m_{1}}\left(x-\lambda_{2}\right)^{m_{2}}$$
where $\lambda_{1} \neq \lambda_{2}$. What are $\operatorname{dim} K_{A}$ and $\operatorname{dim} L_{A} ?$