Is there any order to the symbolic eigenvalues of a matrix returned by the command Eigenvalues[...]
?
While numerical eigenvalues are listed in descending order (i.e., $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_N $), this is not the case for functional eigenvalues (see two examples below). The examples suggest that the ordering may be the reverse, although this cannot be positively inferred from two examples. Do functional eigenvalues have the order $\lambda_1 \le \lambda_2 \le \cdots \lambda_N$, another ordering (based on eigenvectors, for instance), or no ordering at all?
Examples showing that functional eigenvalues are not sorted in descending order.
(1) As a trivial example,
Eigenvalues[{{Exp[x], Exp[x]}, {Exp[x], Exp[x]}}]
returns $\left\{0,2 e^x\right\}$. For real inputs, the latter is always greater than or equal to the former.
(2) As another example, consider
Eigenvalues[{{Exp[x], Exp[x]}, {Exp[x], Exp[x/2]}}]
which returns
$$\left\{\frac{1}{2} \left(e^{x/2}+e^x-e^{x/2} \sqrt{1-2 e^{x/2}+5 e^x}\right),\frac{1}{2} \left(e^{x/2}+e^x+e^{x/2} \sqrt{1-2 e^{x/2}+5 e^x}\right)\right\}.$$
Again, for real inputs, the latter is always greater than or equal to the former:
Plot[%, {x, -10, 10}]
Sin[x]
andCos[x]
? You'd have to useOrderedQ
, likeSort
does, which would sort them as{Cos[x], Sin[x]}
. But then if you definex
to be some number, the list will not be ordered for around half the possible values ofx
. $\endgroup$