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Matrix representation of a symmetric bilinear form

Like the matrix representation of a bilinear form, it is a matrix, but now the matrix has to be a symmetric matrix.

We can then immediately see that the matrix is symmetric, then so is the form. We have:

B (x, y) = x^{T} M y

But because

B (x, y)

is a scalar, we have:

B (x, y) = B (x, y)^{T}

and:

B (x, y) = B (x, y)^{T} = (x^{T} M y)^{T} = y^{T} M^{T} x = y^{T} M^{T} x = y^{T} M x = B (y, x)

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