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Kronecker (Tensor) Product

A⊗B: each aᵢⱼ replaced by aᵢⱼ·B. (A⊗B) has blocks [aᵢⱼB]. Dimensions: (mp)×(nq) for A m×n, B p×q. (A⊗B)(C⊗D) = (AC)⊗(BD) when dimensions match.

Concept Fundamentals
(mp)×(nq)
Dimensions
[aᵢⱼB]
Block
(AC)⊗(BD)
(A⊗B)(C⊗D)
state spaces
Quantum

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(A⊗B)⊗C = A⊗(B⊗C) (associative). det(A⊗B) = det(A)^q det(B)^p. Eigenvalues: λᵢμⱼ for A,B.

Key quantities
(mp)×(nq)
Dimensions
Key relation
[aᵢⱼB]
Block
Key relation
(AC)⊗(BD)
(A⊗B)(C⊗D)
Key relation
state spaces
Quantum
Key relation

Ready to run the numbers?

Why: Kronecker product appears in quantum mechanics (composite systems), image processing, and structured linear algebra.

How: For each (i,j) of A, place block aᵢⱼ·B in position (i,j). Result is block matrix with A's structure, B as blocks.

(A⊗B)⊗C = A⊗(B⊗C) (associative).det(A⊗B) = det(A)^q det(B)^p.
Sources:Wolfram MathWorldGilbert Strang, Linear Algebra

Run the calculator when you are ready.

Compute Tensor ProductA⊗B (Kronecker product)

About Tensor Products

The tensor product of matrices A and B (denoted A⊗B) creates a block matrix where each element aij of A is multiplied by the entire matrix B. For an m×n matrix A and a p×q matrix B, the result is an (mp)×(nq) matrix.

Example Matrices

Standard Example

Matrix A (2×2)

1
2
3
4

Matrix B (2×2)

0
5
6
7

Identity Matrices

Matrix A (2×2)

1
0
0
1

Matrix B (2×2)

1
0
0
1

Row × Column

Matrix A (1×2)

1
2

Matrix B (2×1)

3
4

Matrix A

Enter the values for your 2×2 matrix:
⟪
⟫
⟨
⟩

Matrix B

Enter the values for your 2×2 matrix:
⟪
⟫
⟨
⟩

Expected Result Size:

4×4

For an 2×2 matrix A and a 2×2 matrix B, the tensor product A⊗B will be a 4×4 matrix.

What is a Tensor Product?

The tensor product (also known as the Kronecker product) is a binary operation that takes two matrices and produces a larger matrix. It represents a mathematical operation that generalizes the outer product of vectors to matrices. The tensor product is fundamental in various areas of mathematics, physics, and quantum computing.

Key Properties:

  • The tensor product is bilinear: (αA + βC) ⊗ B = α(A ⊗ B) + β(C ⊗ B)
  • It is not commutative: A ⊗ B ≠ B ⊗ A (in general)
  • It is associative: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)
  • Distributive with matrix multiplication: (A ⊗ B)(C ⊗ D) = AC ⊗ BD

How to Calculate a Tensor Product

Algorithm:

  1. Start with an empty result matrix of size (mp)×(nq), where A is m×n and B is p×q.
  2. For each element aij in matrix A:
    • Multiply the entire matrix B by aij
    • Place this resulting submatrix in the result matrix at the position corresponding to aij

Example Calculation:

For A = [1 2; 3 4] and B = [0 5; 6 7]:

1. a11 = 1:
1 × [0 5; 6 7] = [0 5; 6 7]
Place in top-left block of result

2. a12 = 2:
2 × [0 5; 6 7] = [0 10; 12 14]
Place in top-right block of result

3. a21 = 3:
3 × [0 5; 6 7] = [0 15; 18 21]
Place in bottom-left block of result

4. a22 = 4:
4 × [0 5; 6 7] = [0 20; 24 28]
Place in bottom-right block of result

Final result:
[0 5 0 10; 6 7 12 14; 0 15 0 20; 18 21 24 28]

Applications

Quantum Computing

In quantum computing, tensor products represent the combination of quantum states. The state space of a composite quantum system is the tensor product of the state spaces of its components, which enables quantum entanglement.

Image Processing

Tensor products are used in image processing for separable convolutions. A 2D filter can be expressed as the tensor product of two 1D filters, significantly reducing computational complexity.

Graph Theory

The adjacency matrix of the tensor product of two graphs is the tensor product of their individual adjacency matrices, useful for analyzing complex network structures.

Differential Equations

Tensor products appear in numerical methods for solving partial differential equations, particularly when using spectral methods on rectangular domains.

Frequently Asked Questions

What is the difference between tensor product and matrix multiplication?

Matrix multiplication (AB) combines matrices by multiplying rows of A with columns of B, requiring the number of columns in A to equal the number of rows in B. Tensor product (A⊗B) creates a block matrix where each element of A multiplies the entire matrix B, with no dimensional constraints. Matrix multiplication results in an m×p matrix (for m×n and n×p matrices), while tensor product yields an (mp)×(nq) matrix (for m×n and p×q matrices).

Is the tensor product commutative?

No, the tensor product is not commutative in general. A⊗B usually differs from B⊗A in both dimensions and values. The dimensions of A⊗B are (m₁p₁)×(n₁q₁) while B⊗A has dimensions (p₁m₁)×(q₁n₁). Even when both matrices are square with the same dimensions, the resulting matrices typically have different element arrangements.

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For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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(A⊗B)ᵀ = Aᵀ⊗Bᵀ

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I⊗A and A⊗I commute

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