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)
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