# Complex-Linear Matrices and C-Linear Transformations ... Tapp, Propostion 2.4 ... ...

#### Peter

##### Well-known member
MHB Site Helper

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. 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?

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

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

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

Hope that helps ...

Peter

Last edited: