| course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
42 |
|
Paper 2, Section II, F |
2016 |
Let $M_{n, n}$ denote the vector space over a field $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Given $B \in M_{n, n}$, consider the two linear transformations $R_{B}, L_{B}: M_{n, n} \rightarrow$ $M_{n, n}$ defined by
$$L_{B}(A)=B A, \quad R_{B}(A)=A B$$
(a) Show that $\operatorname{det} L_{B}=(\operatorname{det} B)^{n}$.
[For parts (b) and (c), you may assume the analogous result $\operatorname{det} R_{B}=(\operatorname{det} B)^{n}$ without proof.]
(b) Now let $F=\mathbb{C}$. For $B \in M_{n, n}$, write $B^{}$ for the conjugate transpose of $B$, i.e., $B^{}:=\bar{B}^{T}$. For $B \in M_{n, n}$, define the linear transformation $M_{B}: M_{n, n} \rightarrow M_{n, n}$ by
$$M_{B}(A)=B A B^{*}$$
Show that $\operatorname{det} M_{B}=|\operatorname{det} B|^{2 n}$.
(c) Again let $F=\mathbb{C}$. Let $W \subseteq M_{n, n}$ be the set of Hermitian matrices. [Note that $W$ is not a vector space over $\mathbb{C}$ but only over $\mathbb{R} .]$ For $B \in M_{n, n}$ and $A \in W$, define $T_{B}(A)=B A B^{*}$. Show that $T_{B}$ is an $\mathbb{R}$-linear operator on $W$, and show that as such,
$$\operatorname{det} T_{B}=|\operatorname{det} B|^{2 n}$$