Let \(X\) be an \(n \times 1\) matrix over \(C\). Under what conditions does \(O(n, C)\) contain a matrix \(M\) whose first column is \(X\)?
Solution
Write \[ \langle Y,Z\rangle=Y^tZ \] for the standard non-degenerate symmetric bilinear form on \(\mathbb C^n\).
1. A necessary condition is \(X^tX=1\).
If \(M\in O(n,\mathbb C)\) has first column \(X\), then Exercise 4 gives \[ X^tX=M_1^tM_1=1. \] Thus every possible first column must satisfy \[ \sum_{j=1}^n x_j^2=1. \]
2. The condition \(X^tX=1\) is sufficient.
Assume \[ X^tX=1. \] Then the line \[ L=\mathbb C X \] is non-degenerate for the form \(\langle\ ,\ \rangle\). Hence \[ \mathbb C^n=L\oplus L^\perp, \] where \[ L^\perp=\{Y:X^tY=0\}. \] The restriction of the form to \(L^\perp\) is non-degenerate: if \(Y\in L^\perp\) is orthogonal to all of \(L^\perp\), then it is orthogonal both to \(L\) and to \(L^\perp\), hence to all of \(\mathbb C^n\), so \(Y=0\).
Over \(\mathbb C\), every non-degenerate symmetric bilinear form admits an orthonormal basis. Thus choose \[ X_2,\ldots,X_n \] in \(L^\perp\) such that \[ X_j^tX_k=\delta_{jk} \] for \(2\le j,k\le n\). Together with \(X_1=X\), these vectors satisfy \[ X_j^tX_k=\delta_{jk} \] for all \(1\le j,k\le n\).
Let \(M\) be the matrix with columns \[ X_1,X_2,\ldots,X_n. \] By Exercise 4, \[ M\in O(n,\mathbb C), \] and its first column is \(X\).
Therefore \[ \boxed{\text{such an }M\text{ exists}\iff X^tX=1.} \]