diff --git a/base/sparse/sparsematrix.jl b/base/sparse/sparsematrix.jl
index cd8f38b0ccec1..5da9f91a516e4 100644
--- a/base/sparse/sparsematrix.jl
+++ b/base/sparse/sparsematrix.jl
@@ -1404,14 +1404,14 @@ sparse(S::UniformScaling, m::Integer, n::Integer=m) = speye_scaled(S.λ, m, n)
 function map!{Tf,N}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC, Bs::Vararg{SparseMatrixCSC,N})
     _checksameshape(C, A, Bs...)
     fofzeros = f(_zeros_eltypes(A, Bs...)...)
-    fpreszeros = fofzeros == zero(fofzeros)
+    fpreszeros = !_broadcast_isnonzero(fofzeros)
     return fpreszeros ? _map_zeropres!(f, C, A, Bs...) :
                         _map_notzeropres!(f, fofzeros, C, A, Bs...)
 end
 function map{Tf,N}(f::Tf, A::SparseMatrixCSC, Bs::Vararg{SparseMatrixCSC,N})
     _checksameshape(A, Bs...)
     fofzeros = f(_zeros_eltypes(A, Bs...)...)
-    fpreszeros = fofzeros == zero(fofzeros)
+    fpreszeros = !_broadcast_isnonzero(fofzeros)
     maxnnzC = fpreszeros ? min(length(A), _sumnnzs(A, Bs...)) : length(A)
     entrytypeC = _broadcast_type(f, A, Bs...)
     indextypeC = _promote_indtype(A, Bs...)
@@ -1429,14 +1429,14 @@ function broadcast!{Tf,N}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC, Bs::Var
     _aresameshape(C, A, Bs...) && return map!(f, C, A, Bs...) # could avoid a second dims check in map
     Base.Broadcast.check_broadcast_indices(indices(C), A, Bs...)
     fofzeros = f(_zeros_eltypes(A, Bs...)...)
-    fpreszeros = fofzeros == zero(fofzeros)
+    fpreszeros = !_broadcast_isnonzero(fofzeros)
     return fpreszeros ? _broadcast_zeropres!(f, C, A, Bs...) :
                         _broadcast_notzeropres!(f, fofzeros, C, A, Bs...)
 end
 function broadcast{Tf,N}(f::Tf, A::SparseMatrixCSC, Bs::Vararg{SparseMatrixCSC,N})
     _aresameshape(A, Bs...) && return map(f, A, Bs...) # could avoid a second dims check in map
     fofzeros = f(_zeros_eltypes(A, Bs...)...)
-    fpreszeros = fofzeros == zero(fofzeros)
+    fpreszeros = !_broadcast_isnonzero(fofzeros)
     indextypeC = _promote_indtype(A, Bs...)
     entrytypeC = _broadcast_type(f, A, Bs...)
     Cm, Cn = Base.to_shape(Base.Broadcast.broadcast_indices(A, Bs...))
@@ -1465,6 +1465,8 @@ _maxnnzfrom(Cm, Cn, A) = nnz(A) * div(Cm, A.m) * div(Cn, A.n)
 @inline _unchecked_maxnnzbcres(Cm, Cn, As) = min(Cm * Cn, sum(_maxnnzfrom_each(Cm, Cn, As)))
 @inline _checked_maxnnzbcres(Cm, Cn, As...) = Cm != 0 && Cn != 0 ? _unchecked_maxnnzbcres(Cm, Cn, As) : 0
 _broadcast_type(f, As...) = Base._promote_op(f, Base.Broadcast.typestuple(As...))
+@inline _broadcast_isnonzero{T<:Number}(x::T) = x != zero(T)
+@inline _broadcast_isnonzero(x) = x != 0
 
 # _map_zeropres!/_map_notzeropres! specialized for a single sparse matrix
 "Stores only the nonzero entries of `map(f, Matrix(A))` in `C`."
@@ -1475,7 +1477,7 @@ function _map_zeropres!{Tf}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC)
         C.colptr[j] = Ck
         for Ak in nzrange(A, j)
             Cx = f(A.nzval[Ak])
-            if Cx != zero(eltype(C))
+            if _broadcast_isnonzero(Cx)
                 Ck > spaceC && (spaceC = _expandstorage!(C, Ck + nnz(A) - (Ak - 1)))
                 C.rowval[Ck] = A.rowval[Ak]
                 C.nzval[Ck] = Cx
@@ -1559,7 +1561,7 @@ function _map_zeropres!{Tf}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC, B::Sp
             # cases are equally or more likely than the Ai < Bi and Bi < Ai cases. Hence
             # the ordering of the conditional chain above differs from that in the
             # corresponding broadcast code (below).
-            if Cx != zero(eltype(C))
+            if _broadcast_isnonzero(Cx)
                 Ck > spaceC && (spaceC = _expandstorage!(C, Ck + (nnz(A) - (Ak - 1)) + (nnz(B) - (Bk - 1))))
                 C.rowval[Ck] = Ci
                 C.nzval[Ck] = Cx
@@ -1660,7 +1662,7 @@ function _broadcast_zeropres!{Tf}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC,
                 # pattern) the Ai < Bi and Bi < Ai cases are equally or more likely than the
                 # Ai == Bi and termination cases. Hence the ordering of the conditional
                 # chain above differs from that in the corresponding map code.
-                if Cx != zero(eltype(C))
+                if _broadcast_isnonzero(Cx)
                     Ck > spaceC && (spaceC = _expandstorage!(C, _unchecked_maxnnzbcres(C.m, C.n, A, B)))
                     C.rowval[Ck] = Ci
                     C.nzval[Ck] = Cx
@@ -1682,7 +1684,7 @@ function _broadcast_zeropres!{Tf}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC,
                 # B's jth column without storing every entry in C's jth column
                 while Bk < stopBk
                     Cx = f(Ax, B.nzval[Bk])
-                    if Cx != zero(eltype(C))
+                    if _broadcast_isnonzero(Cx)
                         Ck > spaceC && (spaceC = _expandstorage!(C, _unchecked_maxnnzbcres(C.m, C.n, A, B)))
                         C.rowval[Ck] = B.rowval[Bk]
                         C.nzval[Ck] = Cx
@@ -1701,7 +1703,7 @@ function _broadcast_zeropres!{Tf}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC,
                     else
                         Cx = fvAzB
                     end
-                    if Cx != zero(eltype(C))
+                    if _broadcast_isnonzero(Cx)
                         Ck > spaceC && (spaceC = _expandstorage!(C, _unchecked_maxnnzbcres(C.m, C.n, A, B)))
                         C.rowval[Ck] = Ci
                         C.nzval[Ck] = Cx
@@ -1723,7 +1725,7 @@ function _broadcast_zeropres!{Tf}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC,
                 # A's jth column without storing every entry in C's jth column
                 while Ak < stopAk
                     Cx = f(A.nzval[Ak], Bx)
-                    if Cx != zero(eltype(C))
+                    if _broadcast_isnonzero(Cx)
                         Ck > spaceC && (spaceC = _expandstorage!(C, _unchecked_maxnnzbcres(C.m, C.n, A, B)))
                         C.rowval[Ck] = A.rowval[Ak]
                         C.nzval[Ck] = Cx
@@ -1742,7 +1744,7 @@ function _broadcast_zeropres!{Tf}(f::Tf, C::SparseMatrixCSC, A::SparseMatrixCSC,
                     else
                         Cx = fzAvB
                     end
-                    if Cx != zero(eltype(C))
+                    if _broadcast_isnonzero(Cx)
                         Ck > spaceC && (spaceC = _expandstorage!(C, _unchecked_maxnnzbcres(C.m, C.n, A, B)))
                         C.rowval[Ck] = Ci
                         C.nzval[Ck] = Cx
@@ -1862,7 +1864,7 @@ function _map_zeropres!{Tf,N}(f::Tf, C::SparseMatrixCSC, As::Vararg{SparseMatrix
             # rows = _updaterow_all(rowsentinel, activerows, rows, ks, stopks, As)
             vals, ks, rows = _fusedupdate_all(rowsentinel, activerow, rows, ks, stopks, As)
             Cx = f(vals...)
-            if Cx != zero(eltype(C))
+            if _broadcast_isnonzero(Cx)
                 Ck > spaceC && (spaceC = _expandstorage!(C, Ck + min(length(C), _sumnnzs(As...)) - (sum(ks) - N)))
                 C.rowval[Ck] = activerow
                 C.nzval[Ck] = Cx
@@ -1982,7 +1984,7 @@ function _broadcast_zeropres!{Tf,N}(f::Tf, C::SparseMatrixCSC, As::Vararg{Sparse
                 # rows = _updaterow_all(rowsentinel, activerows, rows, ks, stopks, As)
                 args, ks, rows = _fusedupdatebc_all(rowsentinel, activerow, rows, defargs, ks, stopks, As)
                 Cx = f(args...)
-                if Cx != zero(eltype(C))
+                if _broadcast_isnonzero(Cx)
                     Ck > spaceC && (spaceC = _expandstorage!(C, _unchecked_maxnnzbcres(C.m, C.n, As)))
                     C.rowval[Ck] = activerow
                     C.nzval[Ck] = Cx
@@ -2003,7 +2005,7 @@ function _broadcast_zeropres!{Tf,N}(f::Tf, C::SparseMatrixCSC, As::Vararg{Sparse
                 else
                     Cx = defaultCx
                 end
-                if Cx != zero(eltype(C))
+                if _broadcast_isnonzero(Cx)
                     Ck > spaceC && (spaceC = _expandstorage!(C, _unchecked_maxnnzbcres(C.m, C.n, As)))
                     C.rowval[Ck] = Ci
                     C.nzval[Ck] = Cx
diff --git a/test/sparse/sparse.jl b/test/sparse/sparse.jl
index c07a899db1ab5..b4b247253f216 100644
--- a/test/sparse/sparse.jl
+++ b/test/sparse/sparse.jl
@@ -1825,3 +1825,14 @@ let
     @test_throws DimensionMismatch broadcast(+, A, B, speye(N))
     @test_throws DimensionMismatch broadcast!(+, X, A, B, speye(N))
 end
+
+# Test that sparse map and broadcast succeed where the output eltype isn't a
+# concrete subtype of Number. (Issue #19561.)
+let
+    intoneorfloatzero(x) = x != 0.0 ? Int(1) : Float64(x)
+    stringorfloatzero(x) = x != 0.0 ? "Hello" : Float64(x)
+    @test map(intoneorfloatzero, speye(4)) == sparse(map(intoneorfloatzero, eye(4)))
+    @test map(stringorfloatzero, speye(4)) == sparse(map(stringorfloatzero, eye(4)))
+    @test broadcast(intoneorfloatzero, speye(4)) == sparse(broadcast(intoneorfloatzero, eye(4)))
+    @test broadcast(stringorfloatzero, speye(4)) == sparse(broadcast(stringorfloatzero, eye(4)))
+end