I wrote down this code in mathematica:

A = {{1, 2}, {2, 1}};
A // MatrixForm
L = Eigenvalues[A]
V = Eigenvectors[A]

Do[If[L[[i]] < 0, {L[[i]] = -L[[i]], V[[i]] = -V[[i]]}], {i, 
  Length[L]}]

L
V

And it simply gives the correct outputs:

Out:{{1,2},{2,1}}
Out:{3, -1}
Out:{{1, 1}, {-1, 1}}
Out:{3, 1}
Out:{{1, 1}, {1, -1}}

But if I write the matrix as a function of some variable and do the same, it's not giving the same answer:

In: A[a_] = {{a, 2}, {2, a}};

(*Compute eigenvalues and eigenvectors*)
S[a_] = Eigenvalues[A[a]];
T[a_] = Eigenvectors[A[a]];

(*Loop to modify eigenvalues and eigenvectors*)
Do[If[S[a][[i]] < 0, {S[a][[i]] = -S[a][[i]], 
   T[a][[i]] = -T[a][[i]]}], {i, Length[S[a]]}]

S[1]
T[1]


Out:{-1, 3}
Out: {{-1, 1}, {1, 1}}

Can anyone resolve the issue?