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I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.
I am currently focussed on and studying Section 1 in Chapter2, namely:
"1. Complex Matrices as Real Matrices".I need help in fully understanding what Tapp is saying in this section regarding the function
$$ \rho_n \ : \ M_n ( \mathbb{C} ) \rightarrow M_{2n} ( \mathbb{R} ) $$Section 1, Chapter2 reads as follows:
View attachment 3990
View attachment 3991I am having trouble fully understanding how the function:
$$ \rho_n \ : \ M_n ( \mathbb{C} ) \rightarrow M_{2n} ( \mathbb{R} $$
relates to $$f_n$$ and $$R_{ \rho_n (A) }$$ ...
For example if
$$ \rho_2 \begin{pmatrix} a+bi & c+di \\ e + fi & h+ji \end{pmatrix}
= \begin{pmatrix} a & b & c & d \\ -b & a & -d & c \\ e & f & h & j \\ -f & e & -j & h \end{pmatrix}$$
My question is what is
$$f_n$$ and $$R_{ \rho_n (A) }$$
in this case, and how exactly do these expressions relate to $$\rho_n$$ ... ...?
Hope someone can help?
Now in the above the linear transformation $$R_A$$ is mentioned ... ... $$R_A$$ is defined in Tapp Ch. 1, Section 5 as follows:
https://www.physicsforums.com/attachments/3992
View attachment 3993
I am currently focussed on and studying Section 1 in Chapter2, namely:
"1. Complex Matrices as Real Matrices".I need help in fully understanding what Tapp is saying in this section regarding the function
$$ \rho_n \ : \ M_n ( \mathbb{C} ) \rightarrow M_{2n} ( \mathbb{R} ) $$Section 1, Chapter2 reads as follows:
View attachment 3990
View attachment 3991I am having trouble fully understanding how the function:
$$ \rho_n \ : \ M_n ( \mathbb{C} ) \rightarrow M_{2n} ( \mathbb{R} $$
relates to $$f_n$$ and $$R_{ \rho_n (A) }$$ ...
For example if
$$ \rho_2 \begin{pmatrix} a+bi & c+di \\ e + fi & h+ji \end{pmatrix}
= \begin{pmatrix} a & b & c & d \\ -b & a & -d & c \\ e & f & h & j \\ -f & e & -j & h \end{pmatrix}$$
My question is what is
$$f_n$$ and $$R_{ \rho_n (A) }$$
in this case, and how exactly do these expressions relate to $$\rho_n$$ ... ...?
Hope someone can help?
Now in the above the linear transformation $$R_A$$ is mentioned ... ... $$R_A$$ is defined in Tapp Ch. 1, Section 5 as follows:
https://www.physicsforums.com/attachments/3992
View attachment 3993
Last edited: