Let $A$ be the complex matrix representing a transformation of the vector space $C^2$ of 2-tuples over the complex numbers into itself, relative to the natural ordered basis {(1,0),(0,1)}.
Let $A_R$ be a real 4×4 matrix representing the same transformation on the vector space of 2-tuples of complex numbers over the field of real numbers.
Part 1 What is $A_R$ relative to the ordered basis {$(1,0),(0,1),(i,0),(0,i)$} in terms of the entries of A?
Part 2 If $A$ is diagonal, what is $A_R$ relative to the ordered basis {$(1,0),(i,0),(0,1),(0,i)$} in terms of the entries of A?
Part 3 How can the trace and determinant of $A_R$ be computed from the trace and determinant of A?
Part 4 If $A$ and $B$ are similar 2×2 complex matrices, $A_R$ and $B_R$ defined as above relative to the same basis, then are $A_R$ and $B_R$ necessarily similar real matrices? If $A_R$ and $B_R$ are similar 4×4 real matrices, are A and B then necessarily similar complex matrices?
My work:
I have solved parts 1 and 2. Pretty straightforward, after getting used to going from a $C$-basis to an $R$-basis and noting that $dimC^2$ = 2 (over $C$), while $dimC^2$ = 4 (over $R$). But I am stuck on part 3, and would welcome any hints for both parts 3 and 4.
I am guessing for part 3, I need to make some observation about the eigenvalues of the 2x2 complex matrix, compared with the eigenvalues of the 4x4 real matrix (of the same transformation.) I'm wondering whether there is a similarity invariant argument to be made here...
Thanks,