How Do Complex-Linear Matrices Relate to Real Matrices in Proposition 2.5?

[Math Amateur]

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:
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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 ...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
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Hope that helps ...

Peter

 

[jvicens]

Dear Peter,

Thank you for your questions regarding Proposition 2.5 in Kristopher Tapp's book. I am happy to assist you in fully understanding this proposition.

In response to your first question, we can show that there exists a matrix A \in M_n ( \mathbb{C} ) for which the given diagram commutes by using the fact that B \in M_{2n} ( \mathbb{R} ) is complex-linear. This means that B is a linear transformation that preserves the complex structure of \mathbb{C}^n. In other words, B maps complex numbers to complex numbers. Now, since B is a linear transformation, we can represent it as a matrix in the standard basis of \mathbb{C}^n. This matrix will have complex entries, but we can also view it as a real matrix in M_{2n} ( \mathbb{R} ) by considering the real and imaginary parts of each complex entry as separate entries in the matrix. Therefore, there exists a matrix A \in M_n ( \mathbb{C} ) such that B = R_A, where R_A is the matrix representation of A in the standard basis of \mathbb{C}^n. This is how we know that the given diagram commutes.

As for your second question, we can see that R_{ \rho ( -A ) } = R_{-B} by considering the definition of R_A and the fact that B = R_A. R_A is defined as the matrix representation of the linear transformation T_A : \mathbb{C}^n \to \mathbb{C}^n in the standard basis of \mathbb{C}^n. Therefore, R_A is the matrix representation of T_A with respect to the standard basis. Similarly, R_{-B} is the matrix representation of the linear transformation T_{-B} : \mathbb{C}^n \to \mathbb{C}^n in the standard basis. Since B = R_A, we can substitute B for R_A in T_{-B} to get T_{-B} = T_{-R_A}. Then, using the properties of linear transformations, we can show that T_{-R_A} = T_{ \rho ( -A ) }. Therefore, R_{ \rho ( -A ) } = R_{-B}.