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What are the eigenvalues of the linear operator

$T \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} = \begin{bmatrix} 2b & ia+c \\ -3d & i(a-b) \\ \end{bmatrix}$

on the space of two by two complex matrices? I cannot find any by hand or by checking with Mathematica. There may not be any, but I am skeptical of this because then the question I am trying to answer becomes trivial.

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A linear operator on a finite-dimensional complex vector space must have an eigenvalue, by e.g. the Fundamental Theorem of Algebra.

If we use the basis $$ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} , \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} , \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} , \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} $$ for the vector space of $2 \times 2$ complex matrices, the map $T$ has matrix $$ \begin{pmatrix} 0 & 2 & 0 & 0 \\ i & 0 & 1 & 0 \\ 0 & 0 & 0 & -3 \\ i & -i & 0 & 0 \end{pmatrix} $$ We then find that the characteristic polynomial of this matrix (and hence of T) is $$ t^4 - 2i t^2 -3i t + 6i $$ This has four rather nasty complex roots.

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You can just write the transformation as a 4 by 4 matrix and then compute the eigenvalues. Wolframalpha gives 4 distinct complex eigenvalues:

https://www.wolframalpha.com/input/?i=eigenvalues+of+%5B%5B0%2C2%2C0%2C0%5D%2C%5Bi%2C0%2C1%2C0%5D%2C%5B0%2C0%2C0%2C3%5D%2C%5Bi%2C-i%2C0%2C0%5D%5D

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