 Column orthogonality

For a finite group, the sum of products of irreducible character values is zero when $g,h$ are not conjugate, and equals $|C_G(g)|$ when $g,h$ are conjugate. To prove this from row orthogonality relations for a character table, choose representatives $g_j$ of its conjugacy classes and form $U_{ij}=\sqrt{|g_j^G|/|G|}{\chi }_i(g_j)$. Row orthogonality says $U{U}^{\ast }=I$. There are equally many irreducible characters and conjugacy classes, so this is a square unitary matrix and $U^{\ast }U=I$ as well. Its column equations give the result, using $|G|/|g_j^G|=|C_G(g_j)|$.