Creating lower triangle elements by conjugating of upper triangle elements

Question

I have generated the upper triangular elements of a matrix by some loops. Although, here deals with forming an Upper triangular matrix of a list but we want the lower triangular part elements be the conjugate of upper triangular elements. (w, s and d are reals but can have negative values). The first created matrix is

matbefore = {{s, I w - 1, 0, w - 1, I w, I d}, {0, 1 - s, w, 
0, w, I d - 1}, {0, 0, 1 + s, w I,0, w - 2}, {0, 0, 0, 1 + s, d I, 0},  {0, 0, 0 , 0, 1 - s, I d^2}, {0,0, 0, 0, 0, s}};

the matrix which can be represented by

matafter = {{s, I w - 1, 0, w - 1, I w, I d}, {Subscript[a, 21], 1 - s, w, 
0, w, I d - 1}, {Subscript[a, 31], Subscript[a, 32], 1 + s, w I, 
0, w - 2}, {Subscript[a, 41], Subscript[a, 42], Subscript[a, 43], 
1 + s, d I, 0}, {Subscript[a, 51], Subscript[a, 52], Subscript[a, 
53], Subscript[a, 54], 1 - s, I d^2}, {Subscript[a, 61], 
Subscript[a, 62], Subscript[a, 63], Subscript[a, 64], Subscript[a,
 65], s}};

(* I just used Subscript[a,_] to show the lower elements. They are zero before they will be replace with conjugated numbers.*)

For example Subscript[a, 21] must be -1-w I and Subscript[a, 51] must be -w I. Can we use of Hermitian property of the matrix?

Answer 1

You can add the conjugate transpose to the original matrix. If you want the diagonal to remain unchanged, you also need to subtract it out.

FullSimplify[matbefore + Conjugate[Transpose[matbefore]] - 
  DiagonalMatrix@Diagonal[matbefore], {s ∈ Reals, w ∈ Reals, d ∈ Reals}]

Answer 2

I am not exactly sure whether this is what you want, but transposing the matrix and taking the Conjugate of the elements should help

mat /. Subscript[a, _] :> 0 /. e_ :> ComplexExpand[Conjugate[e]] // Transpose

You can use ComplexExpand to tell that all variables s, d and w are real valued in the complex expressions.