 #1
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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:
View attachment 9575
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 complexlinear, 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 complexlinear, 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 ...View attachment 9576
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 ... View attachment 9577
View attachment 9578
View attachment 9579
View attachment 9580
View attachment 9581
Hope that helps ...
Peter
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:
View attachment 9575
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 complexlinear, 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 complexlinear, 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 ...View attachment 9576
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 ... View attachment 9577
View attachment 9578
View attachment 9579
View attachment 9580
View attachment 9581
Hope that helps ...
Peter
Attachments

Tapp  Proposition 2.5 ... .png15.9 KB · Views: 91

Tapp  Defn 1.9 & Defn 1.10 ... .png25.2 KB · Views: 103

Tapp  1  Start of Chapter 2 ... PART 1 ... .png21.8 KB · Views: 86

Tapp  2  Start of Chapter 2 ... PART 2 ... .png33 KB · Views: 94

Tapp  3  Start of Chapter 2 ... PART 3 ... ... .png26.2 KB · Views: 83

Tapp  4  Start of Chapter 2 ... PART 4 ... ... .png28.5 KB · Views: 88

Tapp  5  Start of Chapter 2 ... PART 5 ... ... .png11 KB · Views: 87