Tensor Product Calculator

Calculate Kronecker product A ⊗ B of two matrices with step-by-step solutions

Matrix Input

Matrix A

Matrix B

Tensor Product A ⊗ B (4×4)

5
6
10
12
7
8
14
16
15
18
20
24
21
24
28
32

Trace Properties

tr(A): 5

tr(B): 13

tr(A⊗B): 65

Formula: tr(A⊗B) = tr(A) × tr(B)

Determinant Properties

det(A): -2

det(B): -2

Formula: det(A⊗B) = det(A)^n × det(B)^m

Step-by-Step Calculation

Step 1: Matrix A is 2×2, Matrix B is 2×2

Step 2: Result will be 4×4 (2×2 by 2×2)

Step 3: Computing A ⊗ B using the Kronecker product formula:

Step 4: Block (1,1): a₍1,1₎ × B = 1 × B

Step 5: Block (1,2): a₍1,2₎ × B = 2 × B

Example: 2×2 Tensor Product

Given Matrices

Matrix A = [1 2; 3 4] and Matrix B = [5 6; 7 8]

Result will be 4×4 matrix

Calculation Process

Block (1,1): 1 × B = [5 6; 7 8]

Block (1,2): 2 × B = [10 12; 14 16]

Block (2,1): 3 × B = [15 18; 21 24]

Block (2,2): 4 × B = [20 24; 28 32]

Result: A ⊗ B = [5 6 10 12; 7 8 14 16; 15 18 20 24; 21 24 28 32]

Kronecker Product Properties

Not Commutative

A ⊗ B ≠ B ⊗ A

Order matters!

Associative

(A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)

Grouping doesn't matter

Bilinear

(A+B) ⊗ C = A⊗C + B⊗C

Linear in each argument

Key Formulas

Size Formula

If A is m×n and B is p×q, then A⊗B is (m×p)×(n×q)

Trace Formula

tr(A ⊗ B) = tr(A) × tr(B)

Determinant Formula

det(A ⊗ B) = det(A)^n × det(B)^m

Transpose Formula

(A ⊗ B)^T = A^T ⊗ B^T

Understanding Tensor Products

What is a Tensor Product?

The tensor product (also called Kronecker product) A ⊗ B is a matrix operation that creates a larger matrix by replacing each element of matrix A with that element multiplied by the entire matrix B.

Mathematical Definition

For matrices A (m×n) and B (p×q), the tensor product A ⊗ B is an (mp)×(nq) matrix where each element aij of A is replaced by aij × B.

Block Structure

  • Each element aij becomes a block aijB
  • Blocks are arranged in the same pattern as A
  • Result preserves the structure of both matrices

Applications

  • Quantum mechanics (quantum states)
  • Signal processing (image operations)
  • Statistics (multidimensional arrays)
  • Computer graphics (transformations)
  • Machine learning (feature combinations)

Important Notes

  • ⚠️ Not commutative: A ⊗ B ≠ B ⊗ A in general
  • ✓ Associative: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)
  • ✓ Bilinear: Linear in each argument separately
  • 📏 Size grows quickly: Result can be much larger than inputs