Question

Let V=W+W* be a vector space, being the direct product of the (finite dimensional) vector space W and its dual space W*. Now, let us define a bilinearform B: VxV -> R by

<(a,p), (b,q)> := q(a) + p(b).
Now let us suppose we have both e_1, …, e_n Basis of W and e*_1,….,e*_n Basis of W*.

What is the matrix of this bilinear form?

(I know how these matrices usually look like, but the inner product makes me very confused about the layout of this matrix).

Answer 1

To find the matrix representation of the bilinear form B with respect to the given bases, we need to compute the entries of the matrix M associated with B. The entries of M are defined as
M[i,j] := B((e_i, e*_i), (e_j, e*_j))

For simplicity, let's assume that dim(W) = n and let's write the basis vectors as row vectors. Then we have
e_i = [0 ... 1 ... 0]
|__ i-th position __|

and
e*_i(w) = [x_1 ... x_n] [0 ... 1 ... 0]^T
|__ i-th position __|

where x_i is the i-th coordinate of the dual basis vector e*_i and the superscript T denotes matrix transposition.

Using this notation, we can compute the matrix entries as follows:
M[i,j] = B((e_i, e*_i), (e_j, e*_j))
= (e*_i(e_j) + e*_j(e_i))(e*_i(e_j) + e*_j(e_i))^T
= (e*_i(e_j) + e*_j(e_i))[e*_i(e_j) e*_j(e_i)]
= e*_i(e_j)e*_i^T(e_j) + e*_j(e_i)e*_j^T(e_i)
= [e*_1(e_j) e*_2(e_j) ... e*_n(e_j)][e*_1(e_i) e*_2(e_i) ... e*_n(e_i)]^T
+ [e*_1(e_i) e*_2(e_i) ... e*_n(e_i)][e*_1(e_j) e*_2(e_j) ... e*_n(e_j)]^T

where we have used the definition of B and the fact that e*_i^T(e_j) = delta_ij (Kronecker delta).

Therefore, the matrix M has entries given by
M[i,j] = e*_i(e_j)e*_i^T(e_j) + e*_j(e_i)e*_j^T(e_i)

This gives us the general form of the matrix, where the (i,j)-entry is determined by the values of the dual basis vectors on the corresponding basis vectors. However, without explicit knowledge of the basis vectors and the dual basis vectors, it is not possible to write down the matrix in a more explicit form.