Eigenvalue/eigenvector reordering and/or renormalisation?

Question

I have two functions $f(x,y),g(x,y)$ and a matrix

$M=\begin{pmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y}\\\frac{\partial g}{\partial x} & \frac{\partial g}{\partial y}\end{pmatrix}$.

Because the matrix elements are functions of $x$ and $y$, I expect the eigenvalues of $M$ and the components of the eigenvectors of $M$ to also be functions of $x$ and $y$.

Here's a little bit of code:

f=...;
g=...;
M={{D[f,x],D[f,y]},{D[g,x],D[g,y]}};

(note that I haven't given the explicit expressions for $f$ and $g$ because they're unimportant for the purposes of this question; suffice it to say that they're complicated enough for their partial derivatives to be horrendous).

I can ask Mathematica for the eigenvalues of $M$, and I get a pair of pretty decent (if horrible-looking) answers which are indeed functions of $x$ and $y$. If I attempt to extract the eigenvectors of $M$, however, Mathematica runs for a while and ultimately gives up:

Eigenvectors[M]

Eigenvectors::eivec0: Unable to find all eigenvectors. >>

{{0,0},{0,0}}

I've therefore resorted to graphing the (absolute values of the) components of the eigenvectors for different values of $x$ and $y$, which takes forever but works. Here's my code:

Quiet[
 DensityPlot[
  Abs[Eigenvectors[M][[1]][[1]]],
  {x,x1,x2},{y,y1,y2},
  Exclusions->None,
  ColorFunction->"Rainbow",
  PlotRangePadding->0,
  PlotLabel->Row[{Subscript[v,Subscript[1,1]]}],
  PlotLegends->Automatic
 ]
]

and similarly for the other three components. Here's an example of one of the components:

Now, the Mathematica documentation for Eigenvalues says

If they are numeric, eigenvalues are sorted in order of decreasing absolute value.

and the documentation for Eigenvectors says

Eigenvectors with numeric eigenvalues are sorted in order of decreasing absolute value of their eigenvalues.

and

For exact or symbolic matrices $m$, the eigenvectors are not normalized.

I therefore have four questions:

1. If one eigenvalue has a higher absolute value than the other for certain values of $x$ and $y$ but a lower absolute value than the other for other values of $x$ and $y$, does the order change? In other words, if I ask Mathematica to graph the first eigenvector (i.e. that with the highest absolute value, as per the documentation), do I get one eigenvector for certain values of $x$ and $y$ and the other one for the other values of $x$ and $y$?

2. If the eigenvalues are not normalised, which criterion does Mathematica use to determine which multiples of the normalised eigenvectors to display? (Any linear combination of eigenvectors of a matrix is also an eigenvector of that matrix, so whatever Mathematica returns is $k_1$ times the normalised first eigenvector and $k_2$ times the normalised second eigenvector; how does Mathematica choose the $k_j$? I seem to recall always getting eigenvectors where one of the components equals 1; is this the case? If so, which component is equal to 1?)

3. Here I will use the word "normalise" in a general sense; I know that the eigenvectors are not normalised to 1 (as per the documentation), but they might be normalised to something else (which may or may not be different for each eigenvector; that is relevant for question 2 but not for this question). If the eigenvectors are indeed normalised, does Mathematica renormalise them for each combination of values of $x$ and $y$? (If so, my graphs of the components of the eigenvectors are meaningless except as a gauge of how large the quotient between the two components is.)

4. Regardless of whether the eigenvalues of $M$ are functions of $x$ and $y$ or not, if I call them $e_1$ and $e_2$ (in the order in which Mathematica gives them), is there any way of knowing whether the diagonalised expression for $M$ is $\begin{pmatrix}e_1 & 0\\0 & e_2\end{pmatrix}$ or $\begin{pmatrix}e_2 & 0\\0 & e_1\end{pmatrix}$?

I am aware of this question, but that concerns only the relative ordering of eigenvectors to the ordering of eigenvalues, which is already answered in the documentation (and included in the quotes above). I am also aware of this other question, but that appears to be about intentionally reordering the eigenvalues rather than finding out whether Mathematica does it by default or not.

Thanks for any help.

Answer 1

I've run some more tests and have found the answer to questions 2 and 3. Mathematica does seem to be renormalising with each value of $x$ and $y$. Here's a graph of the norm of one of the eigenvectors:

Quiet[
 Plot3D[
  Sqrt[Abs[Eigenvectors[m][[2]][[1]]]^2+Abs[Eigenvectors[m][[2]][[2]]]^2]-1,
  {x,x1,x2},{y,y1,y2},
  Exclusions->None,
  PlotLabel->Row[{"|",Subscript[v,2],"|"}],
  PlotRangePadding->0
 ]
]

I'll ascribe the variation in the norm (of the order of $10^{-16}$) to machine precision (i.e. the norm of this eigenvector is $1$ throughout the domain). This seems to imply that Mathematica is not only renormalising the eigenvectors at each point, but it is actually giving me the eigenvectors normalised to 1 (contrary to what the documentation says).

Of course, this assumes that the order of the eigenvalues and eigenvectors is preserved regardless of which eigenvalue has a larger absolute value at each point. If it isn't, my whole interpretation of the above graph automatically goes out the window.

Edit:

I've found the answer to question 1 by testing a little further. I built a matrix with eigenvalues whose ordering should change in a very obvious way:

M=DiagonalMatrix[{x-2,2-x}];

Graphing the eigenvalues explicitly (i.e. $x-2$ and $2-x$) yields the following:

Plot[{2-x,x-2},{x,0,4}]

However, graphing the eigenvalues as eigenvalues has a different result:

Plot[{Eigenvalues[M][[1]],Eigenvalues[M][[2]]},{x,0,4}]

It is as I feared: Eigenvalues are reordered at each point. Clearly, then, eigenvectors are also reordered at each point, since the eigenvalue and eigenvector ordering must be consistent at all times (at least according to Mathematica). (This also renders my analysis on the answers to questions 2 and 3 wrong, although the ultimate answers (i.e. Mathematica renormalises to $1$ at every point) is still right because of what J.M. said in the comments.)

Other than having explicit expressions for the eigenvector components (which I do not have, as explained at the beginning of my original post), is there a simple way around this reordering?

Another edit:

Finally, as mentioned in the comment section, I've found a way around the problem posed in question 4, but I imagine it's highly dependent on the type of problem one is trying to solve.

In my case, I can graph $f(x,y)$ and $g(x,y)$ and see in which direction(s) they change most strongly. The eigenvalues and eigenvectors of $M$ are related to the partial derivatives of $f$ and $g$ with respect to $x$ and $y$, so I can use the (qualitative) visual information obtained from the graphs of $f$ and $g$ to "guess" which order the eigenvectors and the corresponding eigenvalues should be in (at least in regions where there is a clear difference between the directions of maximum variation of $f$ and $g$).

This is not a very satisfactory answer to question 4, but it is the only one I have so far.