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Complex-Linear Matrices and C-Linear Transformations ... Tapp, Propostion 2.5 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.

I am currently focused on and studying Section 1 in Chapter2, namely:

"1. Complex Matrices as Real Matrices".


I need help in fully understanding Tapp's Proposition 2.5.

Proposition 2.5 and some comments following it read as follows:



Tapp - Proposition 2.5 ... .png



My questions are as follows:


Question 1

In the above text from Tapp we read the following:

" ... ... Suppose that \(\displaystyle B \in M_{2n} ( \mathbb{R} )\) is complex-linear, so there is a matrix \(\displaystyle A \in M_n ( \mathbb{C} )\) for which the following diagram commutes ... ... "

My question is as follows:

Given that \(\displaystyle B \in M_{2n} ( \mathbb{R} )\) is complex-linear, how, exactly do we know that there exists a matrix \(\displaystyle A \in M_n ( \mathbb{C} )\) for which the given diagram commutes ... ... ?



Question 2

In the above text from Tapp we read the following:

" ... ... so the composition of the three downward arrows on the right must equal \(\displaystyle R_{ \rho ( -A ) } = R_{-B}\) ... ... "


My question is as follows:

Why exactly does \(\displaystyle R_{ \rho ( -A ) } = R_{-B}\) ... ...?



Help will be much appreciated ... ...

Peter


===============================================================================


\(\displaystyle R_A\) is defined in the following text ...


Tapp - Defn 1.9 & Defn 1.10 ... .png



For readers of the above post to understand the definitions, notation and context of the questions it would help for readers to have access to the text at the start of Chapter 2 ... so I am providing that text ... as follows ...


Tapp - 1 - Start of Chapter 2 ... PART 1 ... .png
Tapp - 2 - Start of Chapter 2 ... PART 2 ... .png
Tapp - 3 - Start of Chapter 2 ... PART 3 ... ... .png
Tapp - 4 - Start of Chapter 2 ... PART 4 ... ... .png
Tapp - 5 - Start of Chapter 2 ... PART 5 ... ... .png



Hope that helps ...

Peter