Your matrix isn't positive semi-definite for all values of $m1>0, m2>0, m3>0, k>0, σ>0$.

Proof by contradiction...one of the Eigenvalues is negative for the parameter settings below.

mat = (* your matrix *)

Min@Eigenvalues@(mat /. {m1 -> 1, m2 -> 2, m3 -> 3, k -> 4, σ -> 5}) // N
(* -0.0882134  *)

Your matrix isn't positive semi-definite for all values of $m1>0, m2>0, m3>0, k>0, σ>0$.

Proof by contradiction...one of the Eigenvalues is negative for the parameter settings below.

mat = (* your matrix *)

Min@Eigenvalues@(mat /. {m1 -> 1, m2 -> 2, m3 -> 3, k -> 4, σ -> 5}) // N
(* -0.0882134  *)

Digging a little deeper for positve semidefiniteness, all of the leading principal minors must be non-negative. Testing for the 5th principal minor (the determinant of the upper left $5 \times 5$ submatrix)

Reduce[{Det@mat[[1 ;; 5, 1 ;; 5]] < 0, 
       m1 > 0, m2 > 0, m3 > 0, k > 0, σ > 0}, 
      {m1, m2, m3, k, σ}]

(* m1 > 0 && m2 > 0 && m3 > 0 && k > 0 && σ > 0 *)

So the 5th principal minor is always less than zero when your parameters are greater than zero.