Fix edge cases in SymTridiagonal when ev has an extra element (+, -, iszero, isone, etc.)

[mcognetta]

In SymTridiagonal, ev can have the same number of elements as dv, but the last element in ev is typically ignored. This can, however, cause some issues for things that just check all elements of ev. For example, iszero, isone, etc can all fail due to a particular choice of the final element of ev, even though that element is not used.

This PR fixes the issue for iszero, isone, isdiag, istriu, and istril. See also #41089.

Update:

This bug is also present in some constructors (see JuliaLang/LinearAlgebra.jl#874) and in some binary operations (+/- when one operand is SymTridiagonal and the other being Tridiagonal, Bidiagonal, etc when ev has the same number of elements as ev.

This PR was updated to include fixes for these cases as well.

---

Head:

julia> dv = rand(3)
3-element Vector{Float64}:
 0.26763492657215415
 0.24104518570135736
 0.3485385690926436

julia> ev = zeros(3)
3-element Vector{Float64}:
 0.0
 0.0
 0.0

julia> ev[end] = 1
1

julia> S = SymTridiagonal(dv, ev)
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 0.267635  0.0        ⋅ 
 0.0       0.241045  0.0
  ⋅        0.0       0.348539

julia> isdiag(S)
false

julia> S = SymTridiagonal(ones(3), ev)
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 1.0  0.0   ⋅ 
 0.0  1.0  0.0
  ⋅   0.0  1.0

julia> isone(S)
false

julia> S = SymTridiagonal(zeros(3), ev)
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 0.0  0.0   ⋅ 
 0.0  0.0  0.0
  ⋅   0.0  0.0

julia> iszero(S)
false

This PR:

julia> dv = rand(3)
3-element Vector{Float64}:
 0.6174155214734693
 0.1508335261390339
 0.16092821432071902

julia> ev = zeros(3)
3-element Vector{Float64}:
 0.0
 0.0
 0.0

julia> ev[end] = 1
1

julia> S = SymTridiagonal(dv, ev)
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 0.617416  0.0        ⋅ 
 0.0       0.150834  0.0
  ⋅        0.0       0.160928

julia> isdiag(S)
true

julia> S = SymTridiagonal(ones(3), ev)
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 1.0  0.0   ⋅ 
 0.0  1.0  0.0
  ⋅   0.0  1.0

julia> isone(S)
true

julia> S = SymTridiagonal(zeros(3), ev)
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 0.0  0.0   ⋅ 
 0.0  0.0  0.0
  ⋅   0.0  0.0

julia> iszero(S)
true

[mcognetta]

Seems like there are a few more that need to be added here:

julia> SymTridiagonal(rand(3), rand(2)) + SymTridiagonal(rand(3), rand(3))
ERROR: DimensionMismatch("dimensions must match: a has dims (Base.OneTo(2),), b has dims (Base.OneTo(3),), mismatch at 1")
Stacktrace:
 [1] promote_shape
   @ ./indices.jl:178 [inlined]
 [2] promote_shape
   @ ./indices.jl:169 [inlined]
 [3] +(A::Vector{Float64}, Bs::Vector{Float64})
   @ Base ./arraymath.jl:45
 [4] +(A::SymTridiagonal{Float64, Vector{Float64}}, B::SymTridiagonal{Float64, Vector{Float64}})
   @ LinearAlgebra ~/github/julia-dev/usr/share/julia/stdlib/v1.8/LinearAlgebra/src/tridiag.jl:210
 [5] top-level scope
   @ REPL[2]:1

julia> SymTridiagonal(rand(3), rand(2)) - SymTridiagonal(rand(3), rand(3))
ERROR: DimensionMismatch("dimensions must match: a has dims (Base.OneTo(2),), b has dims (Base.OneTo(3),), mismatch at 1")
Stacktrace:
 [1] promote_shape
   @ ./indices.jl:178 [inlined]
 [2] promote_shape
   @ ./indices.jl:169 [inlined]
 [3] -(A::Vector{Float64}, B::Vector{Float64})
   @ Base ./arraymath.jl:38
 [4] -(A::SymTridiagonal{Float64, Vector{Float64}}, B::SymTridiagonal{Float64, Vector{Float64}})
   @ LinearAlgebra ~/github/julia-dev/usr/share/julia/stdlib/v1.8/LinearAlgebra/src/tridiag.jl:211
 [5] top-level scope
   @ REPL[3]:1

[andreasnoack]

LGTM. There is a small conflict that needs to be resolved.

[mcognetta]

LGTM. There is a small conflict that needs to be resolved.

Should be good to go now. Thanks for the review.