Is There a Standard Order for Eigenvalues in a Matrix?

[g.lemaitre]

Homework Statement

the order of eigenvalues is important, but when you calculate an eigenvalue polynomial i am still not aware of any rule that dictates which eigenvalue comes first and which does not. let me explain what i mean. take the matrix
$$\begin{bmatrix} 3 & -2 \\ 5 & -4 \end{bmatrix}$$
The eigenvalues are -2, and 1
I forget the technical name but when you construct a matrix composed of eigenvalues you can have either
$$\begin{bmatrix} -2 & 0 \\ 0 & 1 \end{bmatrix}$$
or
$$\begin{bmatrix} 1 & 0 \\ 0 & -2 \end{bmatrix}$$
There's a big difference between those two matrices so which one is correct? Up until now it seems that the larger number always occupies the upper left corner but i haven't been paying much attention to it.

 

[Ray Vickson]

Homework Statement

the order of eigenvalues is important, but when you calculate an eigenvalue polynomial i am still not aware of any rule that dictates which eigenvalue comes first and which does not. let me explain what i mean. take the matrix
$$\begin{bmatrix} 3 & -2 \\ 5 & -4 \end{bmatrix}$$
The eigenvalues are -2, and 1
I forget the technical name but when you construct a matrix composed of eigenvalues you can have either
$$\begin{bmatrix} -2 & 0 \\ 0 & 1 \end{bmatrix}$$
or
$$\begin{bmatrix} 1 & 0 \\ 0 & -2 \end{bmatrix}$$
There's a big difference between those two matrices so which one is correct? Up until now it seems that the larger number always occupies the upper left corner but i haven't been paying much attention to it.

There is no rule for ordering eigenvalues, and the order is not important. What IS important is to maintain the same order from the start to the finish of a problem, so if you start with the order -2, 1 you should keep it throughout until your calculations have finished. Of course, some algorithms will assume an eigenvalue order such as $\lambda \leq \lambda_2 \leq \cdots \leq \lambda_n$ in the real-eigenvalue case, or perhaps $|\lambda_1| \leq |\lambda_2| \leq \cdots \leq |\lambda_n|$ for the general case, but that will (or ought to be) specified in the user's manual.

RGV