diff --git a/src/MatrixFields/MatrixFields.jl b/src/MatrixFields/MatrixFields.jl
index 6d2630ec6e..64f06402a2 100644
--- a/src/MatrixFields/MatrixFields.jl
+++ b/src/MatrixFields/MatrixFields.jl
@@ -88,8 +88,7 @@ const ColumnwiseBandMatrixField{V, S} = Fields.Field{
 } where {
     V <: AbstractData{<:BandMatrixRow},
     S <: Union{
-        Spaces.FiniteDifferenceSpace,
-        Spaces.ExtrudedFiniteDifferenceSpace,
+        Spaces.AbstractSpace,
         Operators.PlaceholderSpace, # so that this can exist inside cuda kernels
     },
 }
diff --git a/src/MatrixFields/matrix_shape.jl b/src/MatrixFields/matrix_shape.jl
index 6ebe32a1c8..020d9424f1 100644
--- a/src/MatrixFields/matrix_shape.jl
+++ b/src/MatrixFields/matrix_shape.jl
@@ -3,19 +3,34 @@ struct Square <: AbstractMatrixShape end
 struct FaceToCenter <: AbstractMatrixShape end
 struct CenterToFace <: AbstractMatrixShape end
 
+matrix_shape(matrix_field) = matrix_shape(matrix_field, axes(matrix_field))
+
 """
     matrix_shape(matrix_field, [matrix_space])
 
-Returns either `Square()`, `FaceToCenter()`, or `CenterToFace()`, depending on
-whether the diagonal indices of `matrix_field` are `Int`s or `PlusHalf`s and
-whether `matrix_space` is on cell centers or cell faces. By default,
+Returns the matrix shape for a matrix field defined on the `matrix_space`. By default,
 `matrix_space` is set to `axes(matrix_field)`.
+
+When the matrix_space is a finite difference space (extruded or otherwise): the shape is
+either `Square()`, `FaceToCenter()`, or `CenterToFace()`, depending on
+whether the diagonal indices of `matrix_field` are `Int`s or `PlusHalf`s and
+whether `matrix_space` is on cell centers or cell faces. 
+
+When the matrix_space is a spectral element or point space: only a Square() shape is supported.
 """
-matrix_shape(matrix_field, matrix_space = axes(matrix_field)) = _matrix_shape(
+matrix_shape(matrix_field, matrix_space) = _matrix_shape(
     eltype(outer_diagonals(eltype(matrix_field))),
     matrix_space.staggering,
 )
 
+function matrix_shape(
+    matrix_field,
+    matrix_space::Union{Spaces.AbstractSpectralElementSpace, Spaces.PointSpace},
+)
+    @assert eltype(matrix_field) <: DiagonalMatrixRow
+    Square()
+end
+
 _matrix_shape(::Type{Int}, _) = Square()
 _matrix_shape(::Type{PlusHalf{Int}}, ::Spaces.CellCenter) = FaceToCenter()
 _matrix_shape(::Type{PlusHalf{Int}}, ::Spaces.CellFace) = CenterToFace()
diff --git a/src/MatrixFields/single_field_solver.jl b/src/MatrixFields/single_field_solver.jl
index f3712f28d5..e785c7cee0 100644
--- a/src/MatrixFields/single_field_solver.jl
+++ b/src/MatrixFields/single_field_solver.jl
@@ -98,10 +98,10 @@ end
 
 function _single_field_solve_col!(
     ::ClimaComms.AbstractCPUDevice,
-    cache::Fields.ColumnField,
-    x::Fields.ColumnField,
+    cache,
+    x,
     A,
-    b::Fields.ColumnField,
+    b,
 )
     if A isa Fields.ColumnField
         band_matrix_solve!(
diff --git a/test/MatrixFields/flat_spaces.jl b/test/MatrixFields/flat_spaces.jl
new file mode 100644
index 0000000000..c4ad1a730d
--- /dev/null
+++ b/test/MatrixFields/flat_spaces.jl
@@ -0,0 +1,66 @@
+include("matrix_field_test_utils.jl")
+import ClimaCore.MatrixFields: @name, ⋅
+function point_space(::Type{FT}) where {FT}
+    comms_ctx = ClimaComms.SingletonCommsContext(comms_device)
+    coord = Geometry.ZPoint(FT(0))
+    return Spaces.PointSpace(comms_ctx, coord)
+end
+
+
+function se2d_space(::Type{FT}) where {FT}
+    comms_ctx = ClimaComms.SingletonCommsContext(comms_device)
+    domain_x = Domains.IntervalDomain(
+        Geometry.XPoint(FT(0)),
+        Geometry.XPoint(FT(1));
+        periodic = true,
+    )
+    domain_y = Domains.IntervalDomain(
+        Geometry.YPoint(FT(0)),
+        Geometry.YPoint(FT(1));
+        periodic = true,
+    )
+    plane = Domains.RectangleDomain(domain_x, domain_y)
+
+    mesh = Meshes.RectilinearMesh(plane, 1, 1)
+    grid_topology = Topologies.Topology2D(comms_ctx, mesh)
+    quad = Spaces.Quadratures.GLL{1 + 1}()
+    space = Spaces.SpectralElementSpace2D(grid_topology, quad)
+    return space
+end
+
+
+get_jac_type(
+    space::Union{Spaces.PointSpace, Spaces.SpectralElementSpace2D},
+    FT,
+) = MatrixFields.DiagonalMatrixRow{FT}
+
+function get_j_field(space, FT)
+    jac_type = get_jac_type(space, FT)
+    field = Fields.Field(jac_type, space)
+    fill!(parent(field), 1)
+    return field
+end
+
+implicit_vars = (@name(tmp.v1), @name(tmp.v2))
+for FT in (Float32, Float64)
+    ps = point_space(FT)
+    ses = se2d_space(FT)
+    v1 = Fields.zeros(ps)
+    v2 = Fields.zeros(ses)
+    Y = Fields.FieldVector(; :tmp => (; :v1 => v1, :v2 => v2))
+    implicit_blocks = MatrixFields.unrolled_map(
+        var ->
+            (var, var) =>
+                get_j_field(axes(MatrixFields.get_field(Y, var)), FT),
+        implicit_vars,
+    )
+    matrix = MatrixFields.FieldMatrix(implicit_blocks...)
+    alg = MatrixFields.BlockDiagonalSolve()
+    solver = MatrixFields.FieldMatrixSolver(alg, matrix, Y)
+    b1 = random_field(FT, ps)
+    b2 = random_field(FT, ses)
+    x = similar(Y)
+    b = Fields.FieldVector(; :tmp => (; :v1 => b1, :v2 => b2))
+    MatrixFields.field_matrix_solve!(solver, x, matrix, b)
+    @test x == b
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index f794aa12dd..851558f337 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -81,6 +81,8 @@ UnitTest("MatrixFields - non-scalar broadcasting (1)" ,"MatrixFields/matrix_fiel
 UnitTest("MatrixFields - non-scalar broadcasting (2)" ,"MatrixFields/matrix_fields_broadcasting/test_non_scalar_2.jl"),
 UnitTest("MatrixFields - non-scalar broadcasting (3)" ,"MatrixFields/matrix_fields_broadcasting/test_non_scalar_3.jl"),
 UnitTest("MatrixFields - non-scalar broadcasting (4)" ,"MatrixFields/matrix_fields_broadcasting/test_non_scalar_4.jl"),
+UnitTest("MatrixFields - flat spaces" ,"MatrixFields/flat_spaces.jl"),
+    
 # UnitTest("MatrixFields - matrix field broadcast"   ,"MatrixFields/matrix_field_broadcasting.jl"), # too long
 # UnitTest("MatrixFields - operator matrices"        ,"MatrixFields/operator_matrices.jl"), # too long
 # UnitTest("MatrixFields - field matrix solvers"     ,"MatrixFields/field_matrix_solvers.jl"), # too long