Edit: I've cut some parts to try and focus this question on linear algebra
I'm having a lot of difficulty understanding some parts of this paper, which I'm reading out of personal interest, yet am struggling with it and having difficulty finding resources to understand it. I'd be happy for any help with the below trouble sections if possible, if only some guidance on material to study. I also feel like at some points, I may be overcomplicating issues, but not sure.
The paper I'm referring to is linked here, and the page numbers are the ones written on the pages themselves. All equations referenced in the text I quote are included in the question.
I have two particular trouble spots relating to my understanding of linear algebra (I believe at least, although I could be missing something else), which then go into the third question about expansions of the same equations in question in the first two parts. I've bolded the core of my questions just to make that clear, the rest is quotes or supporting notes.
First, in section VIIA2 (page 48-49)
Here we have a Liouville Operator, approximated as follows (equation 105):
$\hat{L} = -\frac{1}{2}(\eta-\eta_0)^Tm^{-1}(\eta - \eta_0) + V(\nu)$
This is a second order approximation in $\eta$, and $\nu$ is not approximated in, and $m^{-1}$ is a Hessian. However, $V(\nu) \approx \frac{1}{2}(\eta_0)^Tm^{-1}(\eta_0)$ because this is acting as a maximizer. The value $\eta_0$ is the maximizer of $\hat{L}$ over $\eta$. I don't feel totally comfortable with this part, yet I get the basic idea of the expansion, that this is a Taylor expansion in $\eta$, and that is why the Hessian is there.
My first question is, in the same section, they say that the eigenvalues are inverses of the masses in the system, and put the following equation (equation 107):
$m_{ij}^{-1}(\nu) = -\frac{\partial^2\hat{L}}{\partial\eta_i\partial\eta_j}|_{\nu,\eta_0(\nu)}$
I don't understand how this expression connects with eigenvalues or why there is a negative sign on the right-hand side. This matrix comes up later in question, and so I'd like to know what this means.
Next, on page 49-50, Section VIIA3, they are discussing the implied conserved quantities due to Noether's theorem.
1. "In equation 105, only differences $\eta_j - \eta_l$ arise, so then global shifts of $\eta_i$ is a symmetry of the Hamiltonian (Liouvillian). These shifts affect the diagonal component of $\eta$, $(1/D)\sum_{i=1}^{D}\eta_i$, which appears in equation (98) for $S$ only in the term $\sum_{i=1}^D\eta_id\nu_i/d\tau$."
Where equation (98) is (and $S$ is the action, within the path-integral formulation framework of field theory)
$S = N \int d\tau\bigg(\sum_{i=1}^D\eta_i\frac{d\nu_i}{d\tau} + \hat{L}\bigg) = N\hat{S}$
For this (the quoted part, not the action equation), just some clarification would be nice. It seemed to me that $\eta$ was a vector, not a matrix, so it is not clear how $\eta$ can have a diagonal component.
These two sections come together in Appendix I (page 73, in text). As a note, this part begins with the action, which later is expanded:
$\hat{S} = \int d\tau(\eta\frac{d\nu}{d\tau} + \hat{L})$
Then I'm lost where, at end of page 73, they say
"...we expand $\hat{S}$ as a bilinear form using three different groupings of terms:
$\hat{S} = \int d\tau\bigg[\eta \cdot \dot{\nu} - \frac{1}{2}(\eta - \eta_0)^{T}m^{-1}(\eta - eta_0) + V(\nu)\bigg] $
$ = \int d\tau \bigg[-\frac{1}{2}(\eta - \eta_0 - m_{\perp}\dot{\nu})^Tm^{-1}(\eta - \eta_0 - m_{\perp}\dot{\nu}) + \frac{1}{2}\dot{\nu}^Tm_{\perp}\dot{\nu} + V(\nu) + \eta_0 \cdot \dot{\nu}\bigg]$
$ = \int d\tau\frac{1}{2}\bigg[-(\eta - \eta_0 - m_{\perp}\dot{\nu})^Tm^{-1}(\eta - eta_0 - m_{\perp}\dot{\nu}) + (\dot{\nu} + \eta_0m^{-1})^Tm_{\perp}(\dot{\nu} + m^{-1}\eta_0)\bigg]$
In order to invert the matrix $m^{-1}$, it has first been necessary to remove the zero-eigenvalues associated with shifts of all $\eta_i$ by a constant, mentioned in Sec. VIIA3. The projection of $m^{-1}$ onto the (D-1)-dimensional hyperplane $\sum_{i=1}^D\eta_i = 0$ is positive-definite, and $m_{\perp}$ is its matrix inverse."
For this section, I don't understand how the second two equations have been reached, and what they mean by the "zero-eigenvalue associated with shifts of all $\eta_i$ by a constant" - does shifting by a constant produce some eigenvalue? I really don't know how to think about what they are saying here. Are they laying out the conditions for inverting $m^{-1}$?