Let \[ C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} \] be a \(2 \times 2\) matrix. We inquire when it is possible to find \(2 \times 2\) matrices \(A\) and \(B\) such that \(C = AB - BA\). Prove that such matrices can be found if and only if \(C_{11} + C_{22} = 0\).
Solution
(\(\Rightarrow\)) Necessity of the trace-zero condition:
1. If \(C = AB - BA\), then \(C_{11} + C_{22} = 0\).
1.1. The trace of any \(2 \times 2\) matrix \(M = \begin{bmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{bmatrix}\) is \(\operatorname{Tr}(M) = M_{11} + M_{22}\).
definition of trace.
1.2. The trace is linear: \(\operatorname{Tr}(AB - BA) = \operatorname{Tr}(AB) - \operatorname{Tr}(BA)\).
linearity of trace.
1.3. For any two \(2 \times 2\) matrices \(A\) and \(B\), \(\operatorname{Tr}(AB) = \operatorname{Tr}(BA)\).
\(\operatorname{Tr}(AB) = A_{11}B_{11} + A_{12}B_{21} + A_{21}B_{12} + A_{22}B_{22} = \operatorname{Tr}(BA)\).
1.4. Hence \(\operatorname{Tr}(C) = \operatorname{Tr}(AB) - \operatorname{Tr}(BA) = 0\), so \(C_{11} + C_{22} = 0\).
2 and 3.
(\(\Leftarrow\)) Sufficiency of the trace-zero condition:
2. If \(C_{11} + C_{22} = 0\), then there exist \(2 \times 2\) matrices \(A\) and \(B\) such that \(C = AB - BA\).
2.1. If \(C = 0\), choose \(A = B = 0\); then \(AB - BA = 0 = C\).
\(0 \cdot 0 - 0 \cdot 0 = 0\).
2.2. Suppose \(C \neq 0\) and \(\operatorname{Tr}(C) = 0\).Then \(C\) is not a scalar multiple of the identity matrix \(I\).
if \(C = \lambda I\), then \(\operatorname{Tr}(C) = 2\lambda = 0 \implies \lambda = 0 \implies C = 0\), contradicting \(C \neq 0\).
2.3. Since \(C\) is not a scalar matrix, there exists a vector \(v \in F^2\) such that \(\{v, Cv\}\) is linearly independent.
if \(Cv\) were parallel to \(v\) for all \(v\), \(C\) would be a scalar matrix.
2.4. With respect to the basis \(\mathcal{B} = \{v, Cv\}\), the linear operator \(C\) has matrix \(C' = \begin{bmatrix} 0 & d \\ 1 & 0 \end{bmatrix}\) for some scalar \(d\).
\(C(v) = 0\cdot v + 1\cdot Cv\), and \(\operatorname{Tr}(C) = 0 \implies 0 + C'_{22} = 0 \implies C'_{22} = 0\).
2.5. The matrix \(C'\) is a commutator of the matrices \(A' = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\) and \(B' = \begin{bmatrix} 0 & d \\ -1 & 0 \end{bmatrix}\):\[ A'B' - B'A' = \begin{bmatrix} 0 & d \\ 0 & 0 \end{bmatrix} - \begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & d \\ 1 & 0 \end{bmatrix} = C'. \]
direct matrix multiplication.
2.6. Let \(P\) be the invertible change-of-basis matrix such that \(C = P C' P^{-1}\).
change of basis theorem for linear transformations.
2.7. Setting \(A = P A' P^{-1}\) and \(B = P B' P^{-1}\):\[ AB - BA = P(A'B' - B'A')P^{-1} = P C' P^{-1} = C. \]
\(P(XY - YX)P^{-1} = (PXP^{-1})(PYP^{-1}) - (PYP^{-1})(PXP^{-1})\) applied to \(X = A', Y = B'\).
3. Conclusion: \(2 \times 2\) matrices \(A\) and \(B\) exist with \(C = AB - BA\) if and only if \(C_{11} + C_{22} = 0\).
1 and 2.
Q.E.D.