course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
39 |
|
Paper 1, Section II, E |
2018 |
Define a Jordan block $J_{m}(\lambda)$. What does it mean for a complex $n \times n$ matrix to be in Jordan normal form?
If $A$ is a matrix in Jordan normal form for an endomorphism $\alpha: V \rightarrow V$, prove that
$$\operatorname{dim} \operatorname{Ker}\left((\alpha-\lambda I)^{r}\right)-\operatorname{dim} \operatorname{Ker}\left((\alpha-\lambda I)^{r-1}\right)$$
is the number of Jordan blocks $J_{m}(\lambda)$ of $A$ with $m \geqslant r$.
Find a matrix in Jordan normal form for $J_{m}(\lambda)^{2}$. [Consider all possible values of $\lambda$.]
Find a matrix in Jordan normal form for the complex matrix
$$\left[\begin{array}{cccc}
0 & 0 & 0 & a_{1} \\
0 & 0 & a_{2} & 0 \\
0 & a_{3} & 0 & 0 \\
a_{4} & 0 & 0 & 0
\end{array}\right]$$
assuming it is invertible.