In the entire book, perturbation theory is used as a qualitative tool to rationalise some chemical phenomena. The authors write that
$$\psi_i = \sum_\mu{T_{ji}\psi_j^\circ}\tag{1}$$
The proof of the derivation for $T_{ji}$ involves the following statement:
$$T_{ji} = \left(\mathbf{C}_j^\circ\right)^\mathrm{T}\mathbf{S}^\circ\mathbf{C}_i \tag{2}$$
where
$$\left(\mathbf{C}_j^\circ\right)^\mathrm{T} = \left(C_{1j}^\circ \quad C_{2j}^\circ \quad \cdots \quad C_{mj}^\circ\right)\tag{3}$$
(indexing the coefficients for the jth unperturbed MO), and
$$ \mathbf{C}_i = \begin{pmatrix} C_{1i} \\ C_{2i} \\ \vdots \\ C_{mi} \end{pmatrix} \tag{4} $$
(indexing the coefficients for the $i$th perturbed MO); $\mathbf{S},$ the overlap matrix, is defined as usual.
I'm not quite sure how the aforementioned statement is obtained. Could someone please walk me through how
$$T_{ji} = \sum\sum C_{nj}^\circ S_{nm}C_{mi}?$$
Source: Orbital Interactions in Chemistry, 2nd edition, page 794-802
>
. $\endgroup$