# Creating lower triangle elements by conjugating of upper triangle elements

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?

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}]


• Or just ConjugateTranspose[matbefore] for that part ;=} – ciao Aug 21 '16 at 7:31

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.

• So much thanks, but if we want the whole matrix we should add the previous parts with this new part?1 – Unbelievable Aug 21 '16 at 2:16
• One problem: I just used Subscript[a,_] to show the lower elements. They are zero before they will be replace. – Unbelievable Aug 21 '16 at 2:20