Adjoint/transpose for tridiagonal matrices preserve structure

[jishnub]

After this, adjoint and transpose for Tridiagonal/SymTridiagonal would preserve the structure. This was already the case for matrices of numbers, and this PR extends this to the general case.

julia> m = [1 2; 3 4]
2×2 Matrix{Int64}:
 1  2
 3  4

julia> T = Tridiagonal(fill(m,3), fill(m,4), fill(m,3))
4×4 Tridiagonal{Matrix{Int64}, Vector{Matrix{Int64}}}:
 [1 2; 3 4]  [1 2; 3 4]      ⋅           ⋅     
 [1 2; 3 4]  [1 2; 3 4]  [1 2; 3 4]      ⋅     
     ⋅       [1 2; 3 4]  [1 2; 3 4]  [1 2; 3 4]
     ⋅           ⋅       [1 2; 3 4]  [1 2; 3 4]

julia> T'
4×4 Tridiagonal{Adjoint{Int64, Matrix{Int64}}, Base.ReshapedArray{Adjoint{Int64, Matrix{Int64}}, 1, Adjoint{Adjoint{Int64, Matrix{Int64}}, Vector{Matrix{Int64}}}, Tuple{}}}:
 [1 3; 2 4]  [1 3; 2 4]      ⋅           ⋅     
 [1 3; 2 4]  [1 3; 2 4]  [1 3; 2 4]      ⋅     
     ⋅       [1 3; 2 4]  [1 3; 2 4]  [1 3; 2 4]
     ⋅           ⋅       [1 3; 2 4]  [1 3; 2 4]

julia> S = SymTridiagonal(fill(m, 4), fill(m, 3))
4×4 SymTridiagonal{Matrix{Int64}, Vector{Matrix{Int64}}}:
 [1 2; 2 4]  [1 2; 3 4]      ⋅           ⋅     
 [1 3; 2 4]  [1 2; 2 4]  [1 2; 3 4]      ⋅     
     ⋅       [1 3; 2 4]  [1 2; 2 4]  [1 2; 3 4]
     ⋅           ⋅       [1 3; 2 4]  [1 2; 2 4]

julia> S' isa SymTridiagonal
true

julia> S'' === S
true

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 93.83%. Comparing base (39ee3af) to head (1a486bb).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1370      +/-   ##
==========================================
+ Coverage   93.81%   93.83%   +0.02%     
==========================================
  Files          34       34              
  Lines       15715    15709       -6     
==========================================
- Hits        14743    14741       -2     
+ Misses        972      968       -4     

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[stevengj]

I think this is only correct for SymTridiagonal{<:Number}.

In particular, the off-diagonal elements should be adjoint.(transpose.(S.ev)) or equivalently conj(S.ev). The reason is that S.ev stores the upper diagonal, but when you take the adjoint, the new upper diagonal comes from the adjoint of the lower diagonal, but the lower diagonal of S is transpose.(S.ev).

(Why we allow SymTridiagonal for non-Number elements is beyond me.)