Chapter 1

Matrix Lie Groups

Problem 1

Let $[ \cdot , \cdot ]_{n,k}$ be the symmetric bilinear form on $\mathbb{R}^{n+k}$ defined in (1.5). Let $g$ be the $(n + k) \times (n + k)$ diagonal matrix with the first $n$ diagonal entries equal to one and the last $k$ diagonal entries equal to minus one:

$$g = \begin{pmatrix} I_n & 0 \\ 0 & -I_k \end{pmatrix}.$$

Show that for all $x, y \in \mathbb{R}^{n+k}$,

$$[ x, y ]_{n,k} = \langle x, g y \rangle.$$

Show that a $(n + k) \times (n + k)$ real matrix $A$ belongs to $O(n; k)$ if and only if

$$g A^T g = A^{-1}.$$

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