Prove an analogue of Exercise 17 in Section 10.2 for skew-symmetric forms.
Solution
1. Let \(f\) be a skew-symmetric bilinear form on a finite-dimensional vector space \(V\) over a field of characteristic \(\neq 2\).
setup.
2. If \(f = 0\), the zero form is represented by the zero matrix.
base case.
3. Otherwise, choose \(u, v \in V\) with \(f(u, v) \neq 0\); rescaling, assume \(f(u, v) = 1\).
\(f\) is nonzero, so some pair has nonzero value; scale \(u\) by \(1/f(u,v)\).
4. The subspace \(W = \operatorname{span}\{u, v\}\) is nondegenerate, and \(f|_W\) has matrix \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) in the basis \((u, v)\).
\(f(u,u) = f(v,v) = 0\) (skew-symmetry) and \(f(u,v) = 1\), so \(f(v,u) = -1\).
5. \(V = W \oplus W^\perp\).
\(W\) is nondegenerate, so \(V = W \oplus W^\perp\).
6. By induction on \(\dim V\), \(f|_{W^\perp}\) has a basis in which its matrix is a block diagonal sum of \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) blocks (and possibly a zero block).
5 and induction.
7. Hence there is a basis of \(V\) in which the matrix of \(f\) is\[\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \oplus \cdots \oplus \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \oplus 0,\]i.e. a block diagonal matrix with \(2 \times 2\) hyperbolic blocks \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) and a zero block.
4 and 6.
8. In particular, the rank of \(f\) is even, and \(f\) is nondegenerate iff \(\dim V\) is even and there is no zero block.
7.
9. Q.E.D.
7 and 8.