Preserve structure in scaling triangular matrices by NaN (#55310)

Addresses the `Matrix` cases from
#55296. This restores the
behavior to match that on v1.10, and preserves the structure of the
matrix on scaling by `NaN`. This behavior is consistent with the
strong-zero behavior for other structured matrix types, and the scaling
may be seen to be occurring in the vector space corresponding to the
filled elements.

After this,
```julia
julia> UpperTriangular(rand(2,2)) * NaN
2×2 UpperTriangular{Float64, Matrix{Float64}}:
 NaN    NaN
    ⋅   NaN
```
cc. @mikmoore

stdlib/LinearAlgebra/src/triangular.jl

@@ -702,6 +702,43 @@ function _triscale!(A::LowerOrUnitLowerTriangular, c::Number, B::UnitLowerTriang
     return A
 end
 
+function _trirdiv!(A::UpperTriangular, B::UpperOrUnitUpperTriangular, c::Number)
+    n = checksize1(A, B)
+    for j in 1:n
+        for i in 1:j
+            @inbounds A[i, j] = B[i, j] / c
+        end
+    end
+    return A
+end
+function _trirdiv!(A::LowerTriangular, B::LowerOrUnitLowerTriangular, c::Number)
+    n = checksize1(A, B)
+    for j in 1:n
+        for i in j:n
+            @inbounds A[i, j] = B[i, j] / c
+        end
+    end
+    return A
+end
+function _trildiv!(A::UpperTriangular, c::Number, B::UpperOrUnitUpperTriangular)
+    n = checksize1(A, B)
+    for j in 1:n
+        for i in 1:j
+            @inbounds A[i, j] = c \ B[i, j]
+        end
+    end
+    return A
+end
+function _trildiv!(A::LowerTriangular, c::Number, B::LowerOrUnitLowerTriangular)
+    n = checksize1(A, B)
+    for j in 1:n
+        for i in j:n
+            @inbounds A[i, j] = c \ B[i, j]
+        end
+    end
+    return A
+end
+
 rmul!(A::UpperOrLowerTriangular, c::Number) = @inline _triscale!(A, A, c, MulAddMul())
 lmul!(c::Number, A::UpperOrLowerTriangular) = @inline _triscale!(A, c, A, MulAddMul())
 
@@ -1095,7 +1132,11 @@ for (t, unitt) in ((UpperTriangular, UnitUpperTriangular),
     tstrided = t{<:Any, <:StridedMaybeAdjOrTransMat}
     @eval begin
         (*)(A::$t, x::Number) = $t(A.data*x)
-        (*)(A::$tstrided, x::Number) = A .* x
+        function (*)(A::$tstrided, x::Number)
+            eltype_dest = promote_op(*, eltype(A), typeof(x))
+            dest = $t(similar(parent(A), eltype_dest))
+            _triscale!(dest, x, A, MulAddMul())
+        end
 
         function (*)(A::$unitt, x::Number)
             B = $t(A.data)*x
@@ -1106,7 +1147,11 @@ for (t, unitt) in ((UpperTriangular, UnitUpperTriangular),
         end
 
         (*)(x::Number, A::$t) = $t(x*A.data)
-        (*)(x::Number, A::$tstrided) = x .* A
+        function (*)(x::Number, A::$tstrided)
+            eltype_dest = promote_op(*, typeof(x), eltype(A))
+            dest = $t(similar(parent(A), eltype_dest))
+            _triscale!(dest, x, A, MulAddMul())
+        end
 
         function (*)(x::Number, A::$unitt)
             B = x*$t(A.data)
@@ -1117,7 +1162,11 @@ for (t, unitt) in ((UpperTriangular, UnitUpperTriangular),
         end
 
         (/)(A::$t, x::Number) = $t(A.data/x)
-        (/)(A::$tstrided, x::Number) = A ./ x
+        function (/)(A::$tstrided, x::Number)
+            eltype_dest = promote_op(/, eltype(A), typeof(x))
+            dest = $t(similar(parent(A), eltype_dest))
+            _trirdiv!(dest, A,  x)
+        end
 
         function (/)(A::$unitt, x::Number)
             B = $t(A.data)/x
@@ -1129,7 +1178,11 @@ for (t, unitt) in ((UpperTriangular, UnitUpperTriangular),
         end
 
         (\)(x::Number, A::$t) = $t(x\A.data)
-        (\)(x::Number, A::$tstrided) = x .\ A
+        function (\)(x::Number, A::$tstrided)
+            eltype_dest = promote_op(\, typeof(x), eltype(A))
+            dest = $t(similar(parent(A), eltype_dest))
+            _trildiv!(dest, x, A)
+        end
 
         function (\)(x::Number, A::$unitt)
             B = x\$t(A.data)

stdlib/LinearAlgebra/test/triangular.jl

@@ -1180,4 +1180,18 @@ end
     @test V == Diagonal([1, 1])
 end
 
+@testset "preserve structure in scaling by NaN" begin
+    M = rand(Int8,2,2)
+    for (Ts, TD) in (((UpperTriangular, UnitUpperTriangular), UpperTriangular),
+                    ((LowerTriangular, UnitLowerTriangular), LowerTriangular))
+        for T in Ts
+            U = T(M)
+            for V in (U * NaN, NaN * U, U / NaN, NaN \ U)
+                @test V isa TD{Float64, Matrix{Float64}}
+                @test all(isnan, diag(V))
+            end
+        end
+    end
+end
+
 end # module TestTriangular