- #1

- 926

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- Homework Statement
- As per title

- Relevant Equations
- N/A

I'd like to clarify a few things; the approach is basically just to show that ##\mathrm{GL}(n,\mathbf{C})## is isomorphic to a subgroup of ##\mathrm{GL}(2n,\mathbf{R})## which is a smooth manifold (since ##\mathbf{R} \setminus \{0\}## is an open subset of ##\mathbf{R}##, so its pre-image ##\mathrm{det}^{-1}(\mathbf{R} \setminus \{ 0\}) = \mathrm{GL}(2n,\mathbf{R})## under the continuous determinant map is a smooth manifold.)

The hint is to consider replacing each ##X_{ij} = a_{ij} + ib_{ij}## with a matrix block ##\begin{pmatrix} a_{ij} & -b_{ij} \\ b_{ij} & a_{ij} \end{pmatrix}##. To show this map is a group isomorphism means to show that ##\Phi(XY) = \Phi(X) \Phi(Y)##, right? So I could write (summation implied over repeated suffices)\begin{align*}

(XY)_{ij} = X_{ik} Y_{kj} &= (a_{ik} + ib_{ik})(c_{kj} + id_{kj}) \\

&= a_{ik} c_{kj} - b_{ik} d_{kj} + i(a_{ik} d_{kj} + b_{ik} c_{kj})

\end{align*}I tried to write the affect of the map ##\Phi## on the elements explicitly, i.e. \begin{align*}

\Phi(X)_{ij} =

\begin{cases}

a_{\mathrm{ceil}{\frac{i}{2}} \mathrm{ceil}{\frac{j}{2}}}, & i+j \ \mathrm{even} \\

(-1)^i b_{\mathrm{ceil}{\frac{i}{2}} \mathrm{ceil}{\frac{j}{2}}}, & i+j \ \mathrm{odd}

\end{cases}

\end{align*}but this becomes a mess to work out ##(\Phi(X) \Phi(Y))_{ij}##. I think it is clear, by considering e.g. a ##1 \times 1## matrix ##(a_{11} + ib_{11})##, that it works, but there is surely a better approach?

The hint is to consider replacing each ##X_{ij} = a_{ij} + ib_{ij}## with a matrix block ##\begin{pmatrix} a_{ij} & -b_{ij} \\ b_{ij} & a_{ij} \end{pmatrix}##. To show this map is a group isomorphism means to show that ##\Phi(XY) = \Phi(X) \Phi(Y)##, right? So I could write (summation implied over repeated suffices)\begin{align*}

(XY)_{ij} = X_{ik} Y_{kj} &= (a_{ik} + ib_{ik})(c_{kj} + id_{kj}) \\

&= a_{ik} c_{kj} - b_{ik} d_{kj} + i(a_{ik} d_{kj} + b_{ik} c_{kj})

\end{align*}I tried to write the affect of the map ##\Phi## on the elements explicitly, i.e. \begin{align*}

\Phi(X)_{ij} =

\begin{cases}

a_{\mathrm{ceil}{\frac{i}{2}} \mathrm{ceil}{\frac{j}{2}}}, & i+j \ \mathrm{even} \\

(-1)^i b_{\mathrm{ceil}{\frac{i}{2}} \mathrm{ceil}{\frac{j}{2}}}, & i+j \ \mathrm{odd}

\end{cases}

\end{align*}but this becomes a mess to work out ##(\Phi(X) \Phi(Y))_{ij}##. I think it is clear, by considering e.g. a ##1 \times 1## matrix ##(a_{11} + ib_{11})##, that it works, but there is surely a better approach?

Last edited: