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I am bringing one short question at the first and after that I will presenet the larger problem: answers may be related to use of Ordering or SortBy but I think they are not useful to the second question:

1- I have a list with these elements:

list={1/2 (1 + Sqrt[5]), 1, 1, 1/2 (1 - Sqrt[5]), 0}; 

I want to sort them with

Sort[list]; 

I see the bellow result:

Which is incorrect, because N[1/2 (1 - Sqrt[5])]= -0.6, However I can write Sort[N[list]] but I need to have the exact numbers, not their approximate values (I mean that I need 1/2 (1 - Sqrt[5]) instead of -0.6).

2-If we have a matrix same as here:

matrix = {{1, 0, 1, 0, 0}, {0, 1, 1, 1, 0}, {1, 0, 0, 0, 0}, {1, 0, 0, 1, 1}, {1, 0, 1, 0, 0}}; 

with this code:

{evals, evecs} = Eigensystem[matrix]; sortedvalues = SortBy[Transpose[{evals, evecs}], First]; 

(I can't add Less in order to sort evals and corresponded evecs)
Also, or by

 {\[CurlyEpsilon], \[Psi]} = Eigensystem[matrix]; {\[CurlyEpsilon], \[Psi]} = {\[CurlyEpsilon][[#]],\[Psi][[#]]} &@Ordering[\[CurlyEpsilon]] 

Ordering is not able to sort and recognize the magnitude of 1/2 (1 - Sqrt[5])!!!!

I have a list with these elements:

list={1/2 (1 + Sqrt[5]), 1, 1, 1/2 (1 - Sqrt[5]), 0}; 

I want to sort them with

Sort[list]; 

I see the bellow result:

Which is incorrect, because N[1/2 (1 - Sqrt[5])]= -0.6, However I can write Sort[N[list]] but I need to have the exact numbers, not their approximate values (I mean that I need 1/2 (1 - Sqrt[5]) instead of -0.6).

I have a list with these elements:

list={1/2 (1 + Sqrt[5]), 1, 1, 1/2 (1 - Sqrt[5]), 0}; 

[![enter image description here][1]][1]

I want to sort them with

Sort[list]; 

I see the bellow result:

[![enter image description here][2]][2]

Which is incorrect, because N[1/2 (1 - Sqrt[5])]= -0.6, However I can write Sort[N[list]] but I need to have the exact numbers, not their approximate values (I mean that I need 1/2 (1 - Sqrt[5]) instead of -0.6). [1]: https://i.sstatic.net/ZWoFv.png [2]: https://i.sstatic.net/8WMYN.png

I am bringing one short question at the first and after that I will presenet the larger problem: answers may be related to use of Ordering or SortBy but I think they are not useful to the second question:

1- I have a list with these elements:

list={1/2 (1 + Sqrt[5]), 1, 1, 1/2 (1 - Sqrt[5]), 0}; 

I want to sort them with

Sort[list]; 

I see the bellow result:

Which is incorrect, because N[1/2 (1 - Sqrt[5])]= -0.6, However I can write Sort[N[list]] but I need to have the exact numbers, not their approximate values (I mean that I need 1/2 (1 - Sqrt[5]) instead of -0.6).

2-If we have a matrix same as here:

matrix = {{1, 0, 1, 0, 0}, {0, 1, 1, 1, 0}, {1, 0, 0, 0, 0}, {1, 0, 0, 1, 1}, {1, 0, 1, 0, 0}}; 

with this code:

{evals, evecs} = Eigensystem[matrix]; sortedvalues = SortBy[Transpose[{evals, evecs}], First]; 

(I can't add Less in order to sort evals and corresponded evecs)
Also, or by

 {\[CurlyEpsilon], \[Psi]} = Eigensystem[matrix]; {\[CurlyEpsilon], \[Psi]} = {\[CurlyEpsilon][[#]],\[Psi][[#]]} &@Ordering[\[CurlyEpsilon]] 

Ordering is not able to sort and recognize the magnitude of 1/2 (1 - Sqrt[5])!!!!