diff --git a/.gitignore b/.gitignore
index 22b0fc821..3e6da32ea 100644
--- a/.gitignore
+++ b/.gitignore
@@ -6,6 +6,7 @@
 __pycache__
 .ipynb*
 *Manifest.toml
+*LocalPreferences.toml
 .vscode
 experimental
 refs
diff --git a/Project.toml b/Project.toml
index 65345342b..2b1a4750b 100644
--- a/Project.toml
+++ b/Project.toml
@@ -57,7 +57,7 @@ FiniteDifferences = "0.12"
 GPUArrays = "11.4.1"
 LRUCache = "1.6"
 LinearAlgebra = "1"
-MatrixAlgebraKit = "0.6.13"
+MatrixAlgebraKit = "0.6.14"
 Mooncake = "0.5.27"
 OhMyThreads = "0.8.0"
 PrecompileTools = "1"
diff --git a/docs/src/man/factorizations.md b/docs/src/man/factorizations.md
index da0069730..d1fa593df 100644
--- a/docs/src/man/factorizations.md
+++ b/docs/src/man/factorizations.md
@@ -44,3 +44,20 @@ P ≈ P * P
 t′ = permute(t, ((1,), (2, 3)));
 t′ ≈ t′ * P
 ```
+
+To compute only the `R` or `L` factor with QR/LQ-based `left_orth!` and `right_orth!`, pass a zero-dimensional tensor map as the basis output.
+
+```julia
+V = infimum(fuse(codomain(t)), fuse(domain(t)))
+V0 = zerospace(spacetype(t))
+Q0 = similar(t, V0 ← V0)
+
+R = similar(t, V ← domain(t))
+_, R = left_orth!(copy(t), (Q0, R))
+
+L = similar(t, codomain(t) ← V)
+L, _ = right_orth!(copy(t), (L, Q0))
+```
+
+For a `DiagonalTensorMap`, use an empty `DiagonalTensorMap` as the basis output instead.
+Differentiation of these R/L-only paths is not yet supported.
diff --git a/test/factorizations/ortho.jl b/test/factorizations/ortho.jl
index 170d596a2..05f272d80 100644
--- a/test/factorizations/ortho.jl
+++ b/test/factorizations/ortho.jl
@@ -52,6 +52,19 @@ for V in spacelist
                 @test Q * R ≈ t
                 @test isisometric(Q)
 
+                # Compute only R by passing an empty basis output.
+                V0 = zerospace(spacetype(t))
+                Q0 = t isa DiagonalTensorMap ? DiagonalTensorMap(T[], V0) : similar(t, V0 ← V0)
+                for positive in (false, true), alg in ((; positive), DefaultAlgorithm(; positive))
+                    Q, R = @constinferred left_orth(t; positive)
+                    R0 = similar(R)
+                    Q2, R2 = @constinferred left_orth!(copy(t), (Q0, R0); alg)
+                    @test dim(Q2) == 0
+                    @test space(R2) == space(R)
+                    @test R2 ≈ R
+                    @test Q * R2 ≈ t
+                end
+
                 N = @constinferred qr_null(t)
                 @test isisometric(N)
                 @test norm(N' * t) ≈ 0 atol = 100 * eps(norm(t))
@@ -146,6 +159,19 @@ for V in spacelist
                 @test L * Q ≈ t
                 @test isisometric(Q; side = :right)
 
+                # Compute only L by passing an empty basis output.
+                V0 = zerospace(spacetype(t))
+                Q0 = t isa DiagonalTensorMap ? DiagonalTensorMap(T[], V0) : similar(t, V0 ← V0)
+                for positive in (false, true), alg in ((; positive), DefaultAlgorithm(; positive))
+                    L, Q = @constinferred right_orth(t; positive)
+                    L0 = similar(L)
+                    L2, Q2 = @constinferred right_orth!(copy(t), (L0, Q0); alg)
+                    @test dim(Q2) == 0
+                    @test space(L2) == space(L)
+                    @test L2 ≈ L
+                    @test L2 * Q ≈ t
+                end
+
                 Nᴴ = @constinferred lq_null(t)
                 @test isisometric(Nᴴ; side = :right)
                 @test norm(t * Nᴴ') ≈ 0 atol = 100 * eps(norm(t))