Fix bug and add tests for Bidiagonal (#42354)

Author: kshyatt

stdlib/LinearAlgebra/src/bidiag.jl

@@ -735,7 +735,7 @@ function ldiv!(A::Bidiagonal, b::AbstractVector)
     end
 
     if N == 0
-        return x
+        return b
     end
 
     @inbounds begin

stdlib/LinearAlgebra/test/bidiag.jl

@@ -269,6 +269,11 @@ Random.seed!(1)
                     end
                 end
             end
+            zdv = Vector{elty}(undef, 0)
+            zev = Vector{elty}(undef, 0)
+            zA  = Bidiagonal(zdv, zev, :U)
+            zb  = Vector{elty}(undef, 0)
+            @test ldiv!(zA, zb) === zb
         end
 
         if elty <: BlasReal
@@ -549,6 +554,14 @@ end
             B = Bidiagonal(dv, ev, uplo)
             @test dot(x, B, y) ≈ dot(B'x, y) ≈ dot(x, Matrix(B), y)
         end
+        dv = Vector{elty}(undef, 0)
+        ev = Vector{elty}(undef, 0)
+        x = Vector{elty}(undef, 0)
+        y = Vector{elty}(undef, 0)
+        for uplo in (:U, :L)
+            B = Bidiagonal(dv, ev, uplo)
+            @test dot(x, B, y) ≈ dot(zero(elty), zero(elty), zero(elty))
+        end
     end
 end