Kronecker product: (A ⊗ B)ᵢⱼ block = aᵢⱼ·B. If A is m₁×n₁ and B is m₂×n₂, result is m₁m₂ × n₁n₂. Properties: (A⊗B)(C⊗D) = (AC)⊗(BD). det(A⊗B) = det(A)^n₂ · det(B)^n₁. Eigenvalues: λᵢμⱼ.
How to Use
Select matrices:
- A, B: Input matrices
- A ⊗ B: Result
- Size: Product dimensions
Quantum Computing
Two-qubit states: |ψ⟩ ⊗ |φ⟩. Gate on qubit 1: U ⊗ I. CNOT, SWAP are NOT tensor products (they create entanglement). Tensor product = no entanglement.
Properties
- (A⊗B)ᵀ = Aᵀ⊗Bᵀ
- (A⊗B)⁻¹ = A⁻¹⊗B⁻¹
- tr(A⊗B) = tr(A)·tr(B)
- Not commutative: A⊗B ≠ B⊗A
Step-by-Step Instructions
- 1Select preset matrices.
- 2View A and B.
- 3Compute A ⊗ B.
- 4Check dimensions.
- 5Examine properties.