sorting eigenvectors of a final combined matrix

Question

I have two matrices m1 and m2 as below:

m1 = {{1, 2, 4, 6}, {2, 0, 3, 4}, {1, -1, -4, 4}, {0, 2, -2, -1}};
m2 = {{0, -2, 2, 1}, {1, 0, 2, 1}, {1, -1, -2, 1}, {1, 3, -2, -1}};

these matrices are multiplied by varying real numbers n1 and n2 (n1*m1 and n2*m2). Although I can obtain sorted Eigenvectors of n1*m1 with this command:

storeigenm1 = {};
 Block[{nx = 3, intervals = 0.5}, 
  storeigenm1 = 
   Flatten[Last /@ 
     SortBy[Transpose@MapAt[Orthogonalize, Eigensystem[#*m1], 2], 
       First] & /@ Range[0, nx, intervals], 1]];

When I use storeigenm1//MatrixForm I can see a 36*1 matrix, because at any step of n1 (which is varied in the code from nx=0 to 3 by 0.5 intervals that equals to 9 stages and at any stage there are 4 eigenvectors thus the final results has 36 row). Sorting of Eigenvectors done based on Eigenvalues amounts.

{related to lowest Eigenvalue.......................... Eigenvector of 0*m1
 second ..................... Eigenvector of 0*m1
 third.......................... Eigenvector of 0*m1
 fourth........................... Eigenvector of 0*m1

 first............................Eigenvectot of 0.5*m1
 second............................Eigenvectot of 0.5*m1
 third...........................Eigenvectot of 0.5*m1
 fourth............................Eigenvectot of 0.5*m1
  .
  .
  .
  .

..................................Eigenvectot of 3*m1}

But the main goal and main question is related to sort Eigenvectors of a combination of m1 and m2 (which multiplied by n1 and n2) which should be as below

{first........................... Eigenvector of 0*m1 + 0*m2
 second........................... Eigenvector of 0*m1 + 0*m2
third........................... Eigenvector of 0*m1 + 0*m2
fourth........................... Eigenvector of 0*m1 + 0*m2

 first...........................Eigenvectot of 0.5*m1 + 0*m2
 second............................Eigenvectot of 0.5*m1 + 0*m2
 third..........................Eigenvectot of 0.5*m1 + 0*m2
 fourth...........................Eigenvectot of 0.5*m1 + 0*m2
 and so on (same as below in which I don't bring all of element and just I    partly write of them)
..................................Eigenvectot of 1*m1 + 0*m2

....
................................ Eigenvector of 0*m1 + 2.5*m2
 .................................Eigenvectot of 0.5*m1 + 2.5*m2
..................................Eigenvectot of 1*m1 + 2.5*m2

....     
..................................Eigenvectot of 2.5*m1 + 4*m2
..................................Eigenvectot of 3*m1 + 4*m2}

m2 multiplied by 4 since the final value of n2 is different of n1.

I would be so glad to know how I change the main command for this goal?

Answer 1

To get the exact same output as your code, where you use Orthogonalize on the eigenvectors, use this

storeigenm1 = Join @@ Table[
    Sort@Orthogonalize@Eigenvectors[n m1]
    , {n, 0, 3, .5}];

This is easier to understand, as it doesn't have the unneeded Transpose, SortBy, First, Last, or MapAt.

But really, Eigenvectors by default returns normalized, orthogonal vectors when the input matrix is numerical (i.e. not composed of integers or other exact numbers like e and π). And there you have the benefit that they are already sorted in order of their eigenvalues (from largest to smallest). Your output is sorted in order of the first element of the eigenvector, rather than by the eigenvalue which seems a better sorting criterion. Below I reverse the sorting, so that they come in increasing order of the eigenvalue (lowest first) which always seemed a better way to me.

So just do this,

storeigenm1 = Join @@ Table[
    Reverse @ Eigenvectors[N @ n m1]
    , {n, 0.0, 3, .5}];

and skip the Orthogonalize altogether. To do the list you ask for, just add another iterator to the Table and then use Flatten[ ,2] on the result.

storeigenm1m2 = Flatten[
   Table[
    Reverse@Eigenvectors[N@(1 m1 + n2 m2)]
    , {n1, 0, 3, .5}
    , {n2, 0, 4, .5}]
   , 2];

Of course you can use the Sort@Orthogonalize instead here.