The Kronecker product builds large structured matrices from small ones. It describes composite systems in quantum computing (two qubits: 2 × 2 ⊗ 2 × 2), 2D image filters and vectorised matrix equations.
How to use the Tensor Product Calculator
- Matrix A size
- A: a₁
- A: b₁
- A: c₁
- A: a₂
- A: b₂
- A: c₂
- A: a₃
- A: b₃
- A: c₃
- Matrix B size
- B: x₁
- B: y₁
- B: z₁
- B: x₂
- B: y₂
- B: z₂
- B: x₃
- B: y₃
- B: z₃
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Each entry of A is replaced by that entry times the whole matrix B.
Tips
- Check the sizes first: matrix products need the inner dimensions to match.
- Enter the matrix row by row; each row of fields is one row of the matrix.
- Integer matrices are shown with exact fractions where the result is rational.
- A determinant of zero means the matrix has no inverse and its columns are dependent.
Common mistakes to avoid
- Multiplying matrices element by element when a matrix product is needed.
- Assuming AB = BA – matrix multiplication is not commutative.
- Mixing up rows and columns when reading a result.
Formula
A ⊗ B = [aᵢⱼ · B] (each entry of A multiplies the whole of B)
Worked example
Inputs — Matrix A size: 2 × 2; A: a₁: 1; A: b₁: 2; A: a₂: 3; A: b₂: 4; Matrix B size: 2 × 2; B: x₁: 0; B: y₁: 5; B: x₂: 6; B: y₂: 7.
Results — A ⊗ B = ⎡ 0 5 0 10 ⎤
⎢ 6 7 12 14 ⎥
⎢ 0 15 0 20 ⎥
⎣ 18 21 24 28 ⎦; Size = 4×4.
Key terms
- Kronecker product
- The tensor product of two matrices.
Frequently asked questions
What is the size of A ⊗ B?
(rows of A × rows of B) by (columns of A × columns of B).
Is the Kronecker product commutative?
No, but A ⊗ B and B ⊗ A are permutation-similar.
What is (A ⊗ B)(C ⊗ D)?
AC ⊗ BD, when the products exist.
Where is it used?
Quantum computing, signal processing and the vec operator in matrix equations.
What formula does the Tensor Product Calculator use?
A ⊗ B = [aᵢⱼ · B] (each entry of A multiplies the whole of B).