This is quoted from Computational Chemistry by Errol J. Lewars' chapter 2's Stationary Points & Normal-Mode Vibrations: ZPE:
\begin{align}\mathbf H &=\begin{bmatrix} \dfrac{\partial ^2 E}{\partial q_1 \partial q_1} & \dfrac{\partial ^2 E}{\partial q_1 \partial q_2} & \dfrac{\partial ^2 E}{\partial q_1 \partial q_3} & \dots & \dfrac{\partial ^2 E}{\partial q_1 \partial q_9} \\ \dfrac{\partial ^2 E}{\partial q_2 \partial q_1} & \dfrac{\partial ^2 E}{\partial q_2 \partial q_2} & \dfrac{\partial ^2 E}{\partial q_2 \partial q_3} & \dots & \dfrac{\partial ^2 E}{\partial q_2 \partial q_9}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial ^2 E}{\partial q_9 \partial q_1} & \dfrac{\partial ^2 E}{\partial q_9 \partial q_2} & \dfrac{\partial ^2 E}{\partial q_9 \partial q_3} & \dots & \dfrac{\partial ^2 E}{\partial q_9 \partial q_9} \end{bmatrix}\\ \\ &=\underset{\mathbf P}{\begin{bmatrix} q_{11} & q_{12} & q_{13} & \dots & q_{19} \\ q_{21} & q_{22} & q_{23} & \dots & q_{29} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ q_{91} & q_{92} & q_{93} & \dots & q_{99} \end{bmatrix}} \underset{\mathbf k}{\begin{bmatrix} k_{1} & 0 & 0 & \dots & 0 \\ 0 & k_{2} & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & k_{9} \end{bmatrix}}\mathbf P^{-1}\end{align}
The 9 × 9 Hessian for a tri-atomic molecule (three Cartesian coordinates for each atom) is decomposed by diagonalization into a P matrix whose columns are “direction vectors” for the vibrations whose force constants are given by the $\bf k$ matrix. Actually, columns 1, 2 and 3 of $\bf P$ and the corresponding $k_1,k_2,k_3$ of $\bf k$ refer to translational motion of the molecule (motion of the whole molecule from one place to another in space); these three “force constants” are nearly zero. Columns 4, 5 and 6 of $\bf P$ and the corresponding $k_4,k_5,k_6$ of $\bf k$ refer to rotational motion about the three principal axes of rotation, and are also nearly zero. Columns 7, 8 and 9 of $\bf P$ and the corresponding $k_7,k_8,k_9$ of $\bf k$ are the direction vectors and force constants, respectively, for the normal-mode vibrations: and refer to vibrational modes 1, 2 and 3, while the 7th, 8th, and 9th columns of $\bf P$ are composed of the x, y and z components of vectors for motion of the three atoms in mode 1 (column 7), mode 2 (column 8), and mode 3 (column 9).
I couldn't conceive what actually $\bf P$ implies; it is, of course, the eigenvector matrix of the Hessian $\bf H $ but what does it represent?
Lewars wrote that columns of $\bf P$ are actually direction vectors; now what did he mean by direction vector?
I thought $q$ represents the geometric coordinate; I got $k$ is the force-constant; but what are those series of $q$ s in $\mathbf P\;?$
Are the $q$ s in $\bf P$ coordinates of the atoms in molecules? If so, aren't they always changing since they are vibrating, rotating...?
Can anyone please explain this to me what those $q$ s actually represent? What did Lewars want to mean by direction vectors?