diff --git a/include/triton/Tools/LinearLayout.h b/include/triton/Tools/LinearLayout.h
index 37a3d3de90de..595f0e583c2b 100644
--- a/include/triton/Tools/LinearLayout.h
+++ b/include/triton/Tools/LinearLayout.h
@@ -610,6 +610,22 @@ class LinearLayout {
     return *this;
   }
 
+  // Compute a C such that A = B * C if it exists.
+  // In other words, C = B^{-1} * A.
+  // Note that such a C exists iff (every pair of input/output dim of) A is
+  // of the form
+  // [[B, 0],
+  //  [0, C]]
+  // as a matrix, whenever those dimensions are present in B.
+  //
+  // C will always have the same input/output dimensions as A.
+  // When there are dimensions of size 1 there is some ambiguity in the
+  // division, as in `operator*` we treat missing dimensions as dimensions
+  // of size 1 whenever it makes sense to do so. The rule that C has the
+  // same dimensions as A ensures that C is well-defined.
+  friend std::optional<LinearLayout> divideLeft(const LinearLayout &A,
+                                                const LinearLayout &B);
+
   // Returns true if this layout acts trivially (as the identity) on the given
   // dimensions. This means that it's the identity on those dimensions, and it
   // does not map other dimensions onto those or these onto other dimensions.
diff --git a/lib/Tools/LinearLayout.cpp b/lib/Tools/LinearLayout.cpp
index 47255220c346..bd868c7df091 100644
--- a/lib/Tools/LinearLayout.cpp
+++ b/lib/Tools/LinearLayout.cpp
@@ -560,6 +560,80 @@ LinearLayout LinearLayout::concatOuts(const LinearLayout &other) const {
                       /*requiresSurjective=*/false);
 }
 
+std::optional<LinearLayout> divideLeft(const LinearLayout &A,
+                                       const LinearLayout &B) {
+  // Compute a C such that A = B * C if it exists.
+  // Note that such a C exists iff (every pair of input/output dim of) A is of
+  // the form
+  // [[B, 0],
+  //  [0, C]]
+  // as a matrix, whenever those dimensions are present in B.
+  for (StringAttr dim : B.getInDimNames()) {
+    if (!llvm::is_contained(A.getInDimNames(), dim))
+      return std::nullopt;
+  }
+  for (StringAttr dim : B.getOutDimNames()) {
+    if (!llvm::is_contained(A.getOutDimNames(), dim))
+      return std::nullopt;
+  }
+  // Compute candidate C's log-sizes for output dimensions.
+  llvm::MapVector<StringAttr, int32_t> cOutDimSizes;
+  for (StringAttr outDim : A.getOutDimNames()) {
+    int outA = A.getOutDimSizeLog2(outDim);
+    int outB = B.hasOutDim(outDim) ? B.getOutDimSizeLog2(outDim) : 0;
+    int outC = outA - outB;
+    if (outC < 0)
+      return std::nullopt;
+    cOutDimSizes[outDim] = 1 << outC;
+  }
+
+  LinearLayout::BasesT cBases;
+  for (StringAttr inDim : A.getInDimNames()) {
+    int inA = A.getInDimSizeLog2(inDim);
+    int inB = B.hasInDim(inDim) ? B.getInDimSizeLog2(inDim) : 0;
+    int inC = inA - inB;
+    if (inC < 0)
+      return std::nullopt;
+
+    std::vector<std::vector<int32_t>> basesForDim;
+    // Check that A’s first inB entries agree with B.
+    for (int i = 0; i < inB; ++i) {
+      for (StringAttr outDim : A.getOutDimNames()) {
+        int expected = B.hasOutDim(outDim) ? B.getBasis(inDim, i, outDim) : 0;
+        int actual = A.getBasis(inDim, i, outDim);
+        if (actual != expected)
+          return std::nullopt;
+      }
+    }
+
+    // Extract the candidate C bases from the remaining (shifted) entries in A.
+    for (int i = inB; i < inA; ++i) {
+      std::vector<int32_t> candidateBasis;
+      for (StringAttr outDim : llvm::make_first_range(cOutDimSizes)) {
+        int outB = B.hasOutDim(outDim) ? B.getOutDimSizeLog2(outDim) : 0;
+        int v = A.getBasis(inDim, i, outDim);
+
+        // The lower outB bits must be zero.
+        if ((v & ((1 << outB) - 1)) != 0)
+          return std::nullopt;
+        candidateBasis.push_back(v >> outB);
+      }
+      basesForDim.push_back(std::move(candidateBasis));
+    }
+    cBases[inDim] = basesForDim;
+  }
+
+  SmallVector<std::pair<StringAttr, int32_t>> COutDims;
+  for (auto [outDim, outC] : cOutDimSizes) {
+    COutDims.push_back({outDim, outC});
+  }
+  // If the layout A and B are surjective, then C should also be surjective.
+  LinearLayout C(std::move(cBases), COutDims,
+                 /*requireSurjective=*/A.isSurjective() && B.isSurjective());
+  assert(B * C == A);
+  return C;
+}
+
 LinearLayout operator*(LinearLayout inner, LinearLayout outer) {
   // Check that dims common to outer and inner have the same relative order.
   auto inDims = supremum(llvm::to_vector(inner.getInDimNames()),
diff --git a/unittest/Tools/LinearLayoutTest.cpp b/unittest/Tools/LinearLayoutTest.cpp
index 4f89bc9c0ca0..9756eae79fa2 100644
--- a/unittest/Tools/LinearLayoutTest.cpp
+++ b/unittest/Tools/LinearLayoutTest.cpp
@@ -921,6 +921,113 @@ TEST(SupremumTest, ErrorOnInconsistentOrder) {
   ASSERT_DEATH({ supremum(x, y); }, "Supremum does not exist");
 }
 #endif
+
+TEST_F(LinearLayoutTest, DivideLeft_Basic) {
+  // Test division when A = B * C.
+  auto B = LinearLayout::identity1D(8, S("in"), S("out"));
+  auto C = LinearLayout::zeros1D(16, S("in"), S("out"));
+  auto isC = divideLeft(B * C, B);
+  EXPECT_TRUE(isC.has_value());
+  EXPECT_EQ(isC.value(), C);
+
+  auto isB = divideLeft(C * B, C);
+  EXPECT_TRUE(isB.has_value());
+  EXPECT_EQ(isB.value(), B);
+}
+
+TEST_F(LinearLayoutTest, DivideLeft_NonMatchingDims) {
+  // If B contains an extra input dimension not present in A, division should
+  // fail.
+  LinearLayout A = LinearLayout::identity1D(32, S("in"), S("out"));
+  LinearLayout B({{S("in"), {{1}, {2}, {4}, {8}}}, {S("extra"), {{0}}}},
+                 {S("out")});
+  auto candidateOpt = divideLeft(A, B);
+  EXPECT_FALSE(candidateOpt.has_value());
+}
+
+TEST_F(LinearLayoutTest, DivideLeft_Simple) {
+  EXPECT_EQ(divideLeft(LinearLayout::identity1D(8, S("in"), S("out")),
+                       LinearLayout::identity1D(4, S("in"), S("out"))),
+            LinearLayout::identity1D(2, S("in"), S("out")));
+
+  EXPECT_EQ(divideLeft(LinearLayout::identity1D(8, S("in"), S("out")),
+                       LinearLayout::identity1D(8, S("in"), S("out"))),
+            LinearLayout::identity1D(1, S("in"), S("out")));
+}
+
+TEST_F(LinearLayoutTest, DivideLeft_2D) {
+  LinearLayout l1(
+      {
+          {S("in1"), {{1, 1}, {2, 2}, {0, 8}, {0, 4}}},
+          {S("in2"), {{0, 2}, {0, 1}}},
+      },
+      {S("out1"), S("out2")});
+  LinearLayout l2(
+      {
+          {S("in1"), {{1, 1}, {2, 2}}},
+          {S("in2"), {{0, 2}, {0, 1}}},
+      },
+      {S("out1"), S("out2")});
+  LinearLayout l3({{S("in1"), {{0, 2}, {0, 1}}}, {S("in2"), {}}},
+                  {S("out1"), S("out2")});
+  ASSERT_EQ(l2 * l3, l1);
+  ASSERT_EQ(divideLeft(l1, l2).value(), l3);
+}
+
+TEST_F(LinearLayoutTest, DivideLeft_EliminateInDim) {
+  LinearLayout l1(
+      {
+          {S("in2"), {{0, 1}, {1, 0}}},
+          {S("in1"), {{2, 0}, {0, 2}}},
+      },
+      {S("out1"), S("out2")});
+  LinearLayout l2({{S("in2"), {{0, 1}, {1, 0}}}}, {S("out1"), S("out2")});
+  LinearLayout l3({{S("in2"), {}}, {S("in1"), {{1, 0}, {0, 1}}}},
+                  {S("out1"), S("out2")});
+  ASSERT_EQ(l2 * l3, l1);
+  EXPECT_EQ(divideLeft(l1, l2).value(), l3);
+
+  LinearLayout l4({{S("in1"), {{0, 1}, {0, 2}}}, {S("in2"), {}}},
+                  {S("out1"), S("out2")});
+  LinearLayout l5({{S("in1"), {{0, 1}, {0, 2}}}}, {S("out1"), S("out2")});
+  LinearLayout l6({{S("in1"), {}}, {S("in2"), {}}}, {S("out1"), S("out2")});
+  ASSERT_EQ(l5 * l6, l4);
+  EXPECT_EQ(divideLeft(l4, l5).value(), l6);
+
+  LinearLayout l7({{S("in1"), {}}, {S("in2"), {{0, 1}}}, {S("in3"), {}}},
+                  {S("out1"), S("out2")});
+  LinearLayout l8({{S("in2"), {{0, 1}}}}, {S("out1"), S("out2")});
+  LinearLayout l9({{S("in1"), {}}, {S("in2"), {}}, {S("in3"), {}}},
+                  {S("out1"), S("out2")});
+  ASSERT_EQ(l8 * l9, l7);
+  EXPECT_EQ(divideLeft(l7, l8).value(), l9);
+}
+
+TEST_F(LinearLayoutTest, DivideLeft_EliminateOutDim) {
+  LinearLayout l1(
+      {
+          {S("in2"), {{1, 0}, {1, 0}}},
+          {S("in1"), {{2, 0}, {0, 1}}},
+      },
+      {S("out1"), S("out2")});
+  LinearLayout l2({{S("in2"), {{1, 0}, {1, 0}}}}, {S("out1"), S("out2")});
+  LinearLayout l3({{S("in2"), {}}, {S("in1"), {{1, 0}, {0, 1}}}},
+                  {S("out1"), S("out2")});
+  ASSERT_EQ(l2 * l3, l1);
+  EXPECT_EQ(divideLeft(l1, l2).value(), l3);
+
+  LinearLayout l4(
+      {
+          {S("in1"), {{0, 1}, {0, 2}}},
+      },
+      {S("out1"), S("out2")});
+  using BasesArray =
+      ArrayRef<std::pair<StringAttr, std::vector<std::vector<int32_t>>>>;
+  LinearLayout l5(BasesArray{}, {S("out1")});
+  LinearLayout l6({{S("in1"), {{0, 1}, {0, 2}}}}, {S("out1"), S("out2")});
+  ASSERT_EQ(l5 * l6, l4);
+  EXPECT_EQ(divideLeft(l4, l5).value(), l6);
+}
 } // anonymous namespace
 } // namespace mlir::triton