Diagonalization of Hessian H= PkP$^{-1}$: what do the elements of P matrix physically mean? This is quoted from Computational Chemistry by Errol J. Lewars' chapter 2's Stationary Points &amp; Normal-Mode
Vibrations: ZPE:


  \begin{align}\mathbf H &amp;=\begin{bmatrix}
    \dfrac{\partial ^2 E}{\partial q_1 \partial q_1} &amp; \dfrac{\partial ^2 E}{\partial q_1 \partial q_2} &amp; \dfrac{\partial ^2 E}{\partial q_1 \partial q_3} &amp; \dots  &amp; \dfrac{\partial ^2 E}{\partial q_1 \partial q_9} \\
    \dfrac{\partial ^2 E}{\partial q_2 \partial q_1} &amp; \dfrac{\partial ^2 E}{\partial q_2 \partial q_2} &amp; \dfrac{\partial ^2 E}{\partial q_2 \partial q_3} &amp; \dots  &amp; \dfrac{\partial ^2 E}{\partial q_2 \partial q_9}\\
    \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
    \dfrac{\partial ^2 E}{\partial q_9 \partial q_1} &amp; \dfrac{\partial ^2 E}{\partial q_9 \partial q_2} &amp; \dfrac{\partial ^2 E}{\partial q_9 \partial q_3} &amp; \dots  &amp; \dfrac{\partial ^2 E}{\partial q_9 \partial q_9}
\end{bmatrix}\\ \\ &amp;=\underset{\mathbf P}{\begin{bmatrix}
    q_{11} &amp; q_{12} &amp; q_{13} &amp; \dots  &amp; q_{19} \\
    q_{21} &amp; q_{22} &amp; q_{23} &amp; \dots  &amp; q_{29} \\
    \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
    q_{91} &amp; q_{92} &amp; q_{93} &amp; \dots  &amp; q_{99}
\end{bmatrix}} \underset{\mathbf k}{\begin{bmatrix}
    k_{1} &amp; 0 &amp; 0 &amp; \dots  &amp; 0 \\
    0 &amp; k_{2} &amp; 0 &amp; \dots  &amp; 0 \\
    \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
    0 &amp; 0 &amp; 0 &amp; \dots  &amp; 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?
Are the $q$s in $\mathbf{P}$ coordinates of the atoms in molecules? 


Close, but not quite. The $q_{i,j}$ represent the displacements from the equilibrium geometry of the appropriate atom/coordinate $i$, for each mode j. Thus, for example, $q_{5,9}$ corresponds to the displacement from equilibrium of atom $2$ along axis $2$ (typically the $y$-axis) when the molecule is undergoing vibration in mode $9$.


  If so, aren't they always changing since they are vibrating, rotating...?


There are no changes due to translation or rotation. It is customary to "project out" the translational and rotational motion from the Hessian and the geometry.  Qualitatively, this means that we simply ignore the translation and rotation and consider the molecule as "fixed" in its equilibrium configuration (allowing for vibrational 'wiggling'), with that configuration always oriented in the same way relative to the coordinate system.

As for the vibrational motion, though: yes, the $q_{i,j}$ values do technically also 'wiggle' as the molecule vibrates. However, the magnitudes of the perturbations $\delta q_{i,j}$ are in many cases very small, and can be neglected. In those cases where they cannot, the harmonic oscillator approximation to the vibrational motion breaks down and is no longer a useful model for vibration along that mode.

See my answers here and here for some links to literature on situations where this occurs (emphasis on the 'internal rotation' topics, though some of the 'anharmonic' and 'Coriolis' references might be somewhat relevant). Here is a Chem.SE question about nitrogen inversion that might be of interest.  More generally, a Google search for "low-frequency vibrational modes" should turn up lots of interesting reading.