Find all bilinear forms on the space of \(n \times 1\) matrices over \(C\) which are invariant under \(O(n, C)\).
Solution
Every bilinear form on \(V=\mathbb C^n\) has a unique representing matrix \(A\in M_n(\mathbb C)\) such that \[ f(X,Y)=X^tAY. \] By Exercise 9, \(f\) is invariant under \(O(n,\mathbb C)\) if and only if \[ AM=MA \] for every \(M\in O(n,\mathbb C)\). We therefore determine this commutant.
1. A matrix commuting with every orthogonal sign matrix is diagonal.
For each \(k\), let \[ D_k=\operatorname{diag}(1,\ldots,1,-1,1,\ldots,1), \] with the \(-1\) in the \(k\)th position. Since \[ D_k^tD_k=I, \] we have \(D_k\in O(n,\mathbb C)\).
Suppose \(A=(a_{ij})\) commutes with every \(D_k\). For \(i\ne j\), choose \(k=i\). Comparing the \((i,j)\) entries of \[ AD_i=D_iA \] gives \[ a_{ij}= -a_{ij}, \] up to the same conclusion after tracking the two sign factors explicitly; thus \[ 2a_{ij}=0. \] Over \(\mathbb C\), this implies \(a_{ij}=0\). Hence \(A\) is diagonal.
2. A diagonal matrix commuting with every orthogonal permutation matrix is scalar.
Every permutation matrix \(P\) satisfies \[ P^tP=I, \] so \(P\in O(n,\mathbb C)\). Write \[ A=\operatorname{diag}(a_1,\ldots,a_n). \] For any transposition matrix exchanging the \(i\)th and \(j\)th coordinate, commutation \[ AP=PA \] forces \[ a_i=a_j. \] Since \(i,j\) are arbitrary, all diagonal entries are equal. Therefore \[ A=cI \] for some \(c\in\mathbb C\).
3. Scalar multiples of the standard form are invariant.
If \(A=cI\), then for \(M\in O(n,\mathbb C)\), \[ M^tAM=cM^tM=cI=A. \] Hence \[ f(X,Y)=cX^tY \] is invariant under the whole orthogonal group.
Thus the invariant bilinear forms are exactly \[ \boxed{f_c(X,Y)=cX^tY\qquad(c\in\mathbb C).} \]