Proving Tapp's Proposition 2.4: Complex Matrices as Real Matrices

[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 how to prove an assertion related to Tapp's Proposition 2.4.

Proposition 2.4 and some comments following it read as follows:
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In the remarks following Proposition 2.4 we read the following:

" ... ... It (\(\displaystyle F\)) is \(\displaystyle \mathbb{C}\)-linear if and only if \(\displaystyle F(i \cdot X) = i \cdot F(X)\) for all \(\displaystyle X \in \mathbb{C}^n\) ... "My question is as follows ... can someone please demonstrate a proof of the fact that \(\displaystyle F\) is \(\displaystyle \mathbb(C)\)-linear if and only if \(\displaystyle F(i \cdot X) = i \cdot F(X)\) for all \(\displaystyle X \in \mathbb{C}^n\) ... Note that even a indication of the main steps of the proof would help ...Help will be much appreciated ...

Peter ===================================================================================
*** EDIT ***

After a little reflection it appears that " ... \(\displaystyle F\) is \(\displaystyle \mathbb{C}\)-linear \(\displaystyle \Longrightarrow\) \(\displaystyle F(i \cdot X) = i \cdot F(X)\) ... " is immediate as ...

... taking \(\displaystyle c = i\) we have ...

\(\displaystyle F(c \cdot X ) = c \cdot F(X)\) \(\displaystyle \Longrightarrow\) \(\displaystyle F(i \cdot X) = i \cdot F(X)\) for \(\displaystyle c \in \mathbb{C}\)Is that correct?

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Tapp defines \(\displaystyle \rho_n\) and \(\displaystyle f_n\) in the following text ... ... View attachment 9574
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\(\displaystyle R_B\) (actually \(\displaystyle R_A\)) is defined in the following text ...
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Note that Tapp uses \(\displaystyle \mathbb{K}\) to denote one of \(\displaystyle \mathbb{R}, \mathbb{C}\), or \(\displaystyle \mathbb{H}\) ... ... Hope that helps ...

Peter

 

[jvicens]

Dear Peter,

Thank you for reaching out for help with understanding Proposition 2.4 in Chapter 2 of Kristopher Tapp's book on Matrix Groups for Undergraduates. I am happy to assist you in fully understanding this proposition and its proof.

Firstly, let us recall the statement of Proposition 2.4:

Proposition 2.4: Let F: \mathbb{C}^n \rightarrow \mathbb{C}^m be a function. Then, F is \mathbb{C}-linear if and only if F(i \cdot X) = i \cdot F(X) for all X \in \mathbb{C}^n.

To prove this proposition, we will need to use the definition of a \mathbb{C}-linear function. A function F: \mathbb{C}^n \rightarrow \mathbb{C}^m is said to be \mathbb{C}-linear if it satisfies the following properties:

1. F(X+Y) = F(X) + F(Y) for all X, Y \in \mathbb{C}^n
2. F(c \cdot X) = c \cdot F(X) for all X \in \mathbb{C}^n and c \in \mathbb{C}

Now, let us prove the "if" part of the proposition, i.e., assuming F(i \cdot X) = i \cdot F(X) for all X \in \mathbb{C}^n, we need to show that F is \mathbb{C}-linear. To do so, we will use the two properties listed above.

Property 1: To show that F is additive, we will take two arbitrary vectors X, Y \in \mathbb{C}^n and show that F(X+Y) = F(X) + F(Y). By the definition of addition of complex numbers, we have (X+Y)_j = X_j + Y_j for j = 1,2,...,n. Now, using the linearity of F, we have:

F(X+Y) = F((X+Y)_1, (X+Y)_2, ..., (X+Y)_n)
= F(X_1+Y_1, X_2+Y_2, ..., X_n+Y_n)
= (F(X_1+