In BEM, an \(\mathcal{H}\)-matrix \(\mathcal{V}\) is a discretization of a bilinear form \(b_V\). The bilinear form \(b_V\) is induced from a boundary integral operator \(V: X \rightarrow Y\). The boundary integral operator is further determined by a kernel function \(\kappa(x,y)\), which depends on two coordinate variables, \(x\) for the field point and \(y\) for the source point in \(\mathbb{R}^d\). The discretization of the bilinear form \(b_V\) needs to choose two finite dimensional function spaces: domain space \(X_h\) and the dual space \(Y_h'\) of the range space \(Y_h\) of the boundary integral operator. The former is the trial space and the latter is the test space.

Therefore, the symmetry of the \(\mathcal{H}\)-matrix \(\mathcal{V}\) depends on

  1. if the kernel function is symmetric, i.e. \(\kappa(x,y) == \kappa(y,x)\) and
  2. if the two function spaces \(X_h\) and \(Y_h'\) are symmetric, i.e. \(X_h == Y_h'\).

The kernel function symmetry determines the value symmetry and the function space symmetry determines the structure symmetry of the \(\mathcal{H}\)-matrix.