Kronecker Product Calculator
Compute the Kronecker (tensor) product A ⊗ B. If A is m×n and B is p×q, the result is an (mp)×(nq) matrix.
Notes
What is the Kronecker Product?
The Kronecker product A ⊗ B replaces each element A[i][j] with the scaled block A[i][j]·B. The result is a block matrix where each block is a scaled copy of B.
Worked Example
Let A = [[1,2],[3,4]] and B = [[0,5],[6,7]]. Each entry of A scales a copy of B:
a₁₁=1 → 1·B = [[0,5],[6,7]]; a₁₂=2 → 2·B = [[0,10],[12,14]]; a₂₁=3 → 3·B = [[0,15],[18,21]]; a₂₂=4 → 4·B = [[0,20],[24,28]]
A ⊗ B (4×4 result):
| 0 | 5 | 0 | 10 |
| 6 | 7 | 12 | 14 |
| 0 | 15 | 0 | 20 |
| 18 | 21 | 24 | 28 |
- Matrix Calculations Guide — Kronecker Product — In-depth notes on Kronecker Product with worked examples
Frequently Asked Questions
How large is the result of a Kronecker product?
If A is m×n and B is p×q, the result A ⊗ B is (m·p)×(n·q). Two 3×3 matrices produce a 9×9 result.
Is the Kronecker product commutative?
No. A ⊗ B ≠ B ⊗ A in general, though they are related by a permutation matrix.
How does the Kronecker product relate to standard matrix multiplication?
The Kronecker product is a generalization. For matrix equations AXB = C, the vectorization identity vec(AXB) = (B^T ⊗ A) vec(X) is extremely useful.