I am trying to diagonalize a somewhat larger matrix that is dependent on several parameters. Applying, say, Eigenvalues
to that matrix is quite straightforward, but Mathematica apparently hits some polynomial equations that it cannot solve and returns some expressions containing Root
, which are again dependent on said parameters. The expressions can be quite convolved, but for the sake of this qustion, let's assume that Mathematica returns $k - kx + x^5=0$, where $x$ is the free variable and $k$ is one of the parameters.
To proceed, I would like to make use of the knowledge that $k$ is very large, indeed much larger than all other involved parameters. Mathematically, I would like to look at the limit $k\rightarrow\infty$. Looking at the equation $k - kx + x^5=0$, it is apparent that $x_0\rightarrow1$ will hold for any solution as $k\rightarrow\infty$. However, Mathematica does not seem to share that insight:
In= Limit[Root[k - k #1 + #1^5 &, 1], k -> Infinity]
Out= Limit[Root[k - k #1 + #1^5 &, 1], k -> ∞]
Is there any way to make Mathematica realize that it can further simplify this kind of expression? Since everything involved here is polynomial, it should in principle be possible to solve that task by suitable expression replacements, but that sounds very cumbersome and error-prone. So it would be great to find a way other than that approach.
Edit: As Carl hinted in his comment, my question might actually be an instance of the XY problem. So here are some more details. A sample matrix could be the one given below. Most entries are close to the diagonal, but there are also some off-diagonal entries. The determinant is always zero.
$$ \left( \begin{array}{cccccccc} -k & \gamma & 0 & 0 & \delta & 0 & \epsilon & 0 \\ 0 & -\gamma -k & 0 & 0 & 0 & \delta & 0 & \epsilon \\ k & 0 & -\xi & \gamma & 0 & 0 & 0 & 0 \\ 0 & k & 0 & -\gamma -\xi & 0 & 0 & 0 & 0 \\ 0 & 0 & \xi & 0 & -\delta -\theta & \gamma & 0 & 0 \\ 0 & 0 & 0 & \xi & 0 & -\gamma -\delta -\theta & 0 & 0 \\ 0 & 0 & 0 & 0 & \theta & 0 & -\epsilon & \gamma \\ 0 & 0 & 0 & 0 & 0 & \theta & 0 & -\gamma -\epsilon \\ \end{array} \right) $$
The matrix itself describes a system of ODEs and I'm interested in long-term behaviour $t\rightarrow\infty$ as well as in solutions for special starting conditions. Diagonalization is the easiest approach to that, but without doubt not the only one, nor the smartest.
Limit[Root[k - k #1 + #1^5 &, 1], k -> Infinity]
evaluates to1
$\endgroup$ – Bob Hanlon Jan 8 '18 at 16:37Root
object should be produced by just callingEigenvalues
. $\endgroup$ – ranguwud Jan 8 '18 at 16:43