3.II.19B

Let $J_{1}$ denote the $2 \times 2$ matrix $\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right)$. Suppose that $T$ is a $2 \times 2$ uppertriangular real matrix with strictly positive diagonal entries and that $J_{1}^{-1} T J_{1} T^{-1}$ is orthogonal. Verify that $J_{1} T=T J_{1}$.

Prove that any real invertible matrix $A$ has a decomposition $A=B C$, where $B$ is an orthogonal matrix and $C$ is an upper-triangular matrix with strictly positive diagonal entries.

Let $A$ now denote a $2 n \times 2 n$ real matrix, and $A=B C$ be the decomposition of the previous paragraph. Let $K$ denote the $2 n \times 2 n$ matrix with $n$ copies of $J_{1}$ on the diagonal, and zeros elsewhere, and suppose that $K A=A K$. Prove that $K^{-1} C K C^{-1}$ is orthogonal. From this, deduce that the entries of $K^{-1} C K C^{-1}$ are zero, apart from $n$ orthogonal $2 \times 2$ blocks $E_{1}, \ldots, E_{n}$ along the diagonal. Show that each $E_{i}$ has the form $J_{1}{ }^{-1} C_{i} J_{1} C_{i}^{-1}$, for some $2 \times 2$ upper-triangular matrix $C_{i}$ with strictly positive diagonal entries. Deduce that $K C=C K$ and $K B=B K$.

[Hint: The invertible $2 n \times 2 n$ matrices $S$ with $2 \times 2$ blocks $S_{1}, \ldots, S_{n}$ along the diagonal, but with all other entries below the diagonal zero, form a group under matrix multiplication.]

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