Let \(V\) be the space of all \(2 \times 3\) matrices over \(\RR\), and let \(f\) be the bilinear form on \(V\) defined by \(f(X, Y) = \operatorname{trace}(X^tAY)\), where \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}. \] Find the matrix of \(f\) in the ordered basis \[ \{E^{11}, E^{12}, E^{13}, E^{21}, E^{22}, E^{23}\} \] where \(E^{ij}\) is the matrix whose only non-zero entry is a 1 in row \(i\) and column \(j\).
Solution
1. \(f(E^{ij}, E^{kl}) = A_{ik} \delta_{jl}\).
\(X^t A Y\) has \((j,l)\)-entry \(\sum_{r,s} X_{rj} A_{rs} Y_{sl}\); for \(X = E^{ij}\) and \(Y = E^{kl}\), this is \(A_{ik}\) when \(j = l\) and \(0\) otherwise, so its trace is \(A_{ik}\delta_{jl}\).
2. The matrix of \(f\) in the given basis is\[\begin{bmatrix} 1 & 0 & 0 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 2 \\ 3 & 0 & 0 & 4 & 0 & 0 \\ 0 & 3 & 0 & 0 & 4 & 0 \\ 0 & 0 & 3 & 0 & 0 & 4 \end{bmatrix}.\]
1, evaluated for all pairs \((i,j), (k,l)\) in the basis order \(E^{11}, E^{12}, E^{13}, E^{21}, E^{22}, E^{23}\).
3. Q.E.D.
2.