Let \(V\) be the set of all \(2 \times 2\) matrices \(A\) with complex entries which satisfy \(A_{11} + A_{22} = 0\).
Show that \(V\) is a vector space over the field of real numbers, with the usual operations of matrix addition and multiplication of a matrix by a scalar.
Find a basis for this vector space.
Let \(W\) be the set of all matrices \(A\) in \(V\) such that \(A_{21} = -\overline{A}_{12}\) (the bar denotes complex conjugation). Prove that \(W\) is a subspace of \(V\) and find a basis for \(W\).
Solution
1. \(V\) is a real vector space.
The condition \(A_{11}+A_{22}=0\) is preserved under matrix addition and multiplication by real scalars, and the zero matrix satisfies it. Thus \(V\) is a real subspace of \(M_2(\CC)\).
2. A real basis of \(V\) is\[ \begin{bmatrix}1&0\\0&-1\end{bmatrix}, \quad \begin{bmatrix}i&0\\0&-i\end{bmatrix}, \quad \begin{bmatrix}0&1\\0&0\end{bmatrix}, \quad \begin{bmatrix}0&i\\0&0\end{bmatrix}, \quad \begin{bmatrix}0&0\\1&0\end{bmatrix}, \quad \begin{bmatrix}0&0\\i&0\end{bmatrix}. \]
Every \(A\in V\) has the form \[ A=\begin{bmatrix}a&b\\c&-a\end{bmatrix}, \qquad a,b,c\in\CC. \] Writing \(a,b,c\) in real and imaginary parts gives a unique real linear combination of the six displayed matrices. Hence they form a basis and \(\dim_{\RR}V=6\).
3. \(W\) is a real subspace of \(V\).
If \(A,B\in W\), then \[ (A+B)_{21}=A_{21}+B_{21} =-\overline{A_{12}}-\overline{B_{12}} =-\overline{(A+B)_{12}}. \] For \(r\in\RR\), \[ (rA)_{21}=rA_{21}=-r\overline{A_{12}}=-\overline{rA_{12}}. \] Thus the defining condition is preserved.
4. A real basis of \(W\) is\[ \begin{bmatrix}1&0\\0&-1\end{bmatrix}, \quad \begin{bmatrix}i&0\\0&-i\end{bmatrix}, \quad \begin{bmatrix}0&1\\-1&0\end{bmatrix}, \quad \begin{bmatrix}0&i\\i&0\end{bmatrix}. \]
Write \(a=x+iy\) and \(b=u+iv\) with \(x,y,u,v\in\RR\). Since \(c=-\overline b=-u+iv\), every matrix in \(W\) is uniquely \[ x\begin{bmatrix}1&0\\0&-1\end{bmatrix} +y\begin{bmatrix}i&0\\0&-i\end{bmatrix} +u\begin{bmatrix}0&1\\-1&0\end{bmatrix} +v\begin{bmatrix}0&i\\i&0\end{bmatrix}. \] Hence the four matrices form a basis.