Tensor Product Calculator

The Kronecker product (tensor product) A ⊗ B of two matrices A (m × n) and B (p × q) produces an (m·p) × (n·q) block matrix. Each element aᵢⱼ of A multiplies the entire matrix B to form a block in the result.

If A is m × n and B is p × q, the Kronecker product A ⊗ B has dimensions (m·p) × (n·q). The total number of elements is (mn)(pq), the product of the element counts of A and B.

No, the Kronecker product is generally not commutative: A ⊗ B ≠ B ⊗ A. The two products have different dimensions unless both matrices are the same size, and even then they are usually different matrices.

Yes, the Kronecker product is associative: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C). You can compute the product in any grouping without changing the result.

Yes, the Kronecker product distributes over addition: A ⊗ (B + C) = A ⊗ B + A ⊗ C and (A + B) ⊗ C = A ⊗ C + B ⊗ C.

The rank of the Kronecker product equals the product of the ranks: rank(A ⊗ B) = rank(A) × rank(B). This property is useful in analyzing the structure of the result.

For square matrices A (n × n) and B (m × m), det(A ⊗ B) = (det(A))^m × (det(B))^n. The determinant of the Kronecker product is determined by the determinants of the factors.

If λ is an eigenvalue of A with eigenvector x, and μ is an eigenvalue of B with eigenvector y, then λμ is an eigenvalue of A ⊗ B with eigenvector x ⊗ y. The eigenvalues of the Kronecker product are all pairwise products.

The trace of the Kronecker product equals the product of the traces: tr(A ⊗ B) = tr(A) × tr(B). This follows from the definition since the diagonal elements of A ⊗ B are products of diagonal elements of A and B.

The Kronecker product is used in quantum mechanics (tensor products of states), computer graphics (transformations), signal processing (multidimensional arrays), and solving matrix equations (vectorization).

For vectors u (m × 1) and v (n × 1), the Kronecker product u ⊗ v is the outer product uvᵀ, producing an m × n matrix. This is a special case where both inputs are column vectors.

The Kronecker product of two diagonal matrices is also diagonal. The diagonal elements are all pairwise products of the diagonal elements of the input matrices.

If A and B are invertible, then (A ⊗ B)⁻¹ = A⁻¹ ⊗ B⁻¹. The inverse of the Kronecker product is the Kronecker product of the inverses.

The transpose of a Kronecker product equals the Kronecker product of the transposes: (A ⊗ B)ᵀ = Aᵀ ⊗ Bᵀ. This property holds for all matrices.

The Kronecker product is the matrix representation of the abstract tensor product from multilinear algebra. In coordinate form, the Kronecker product gives the matrix that represents the linear map on the tensor product space.

Computing the Kronecker product requires O(m × n × p × q) operations for matrices of the given sizes, since each of the mn elements of A must be multiplied by all pq elements of B.

A ⊗ I_n is a block diagonal matrix where each block is aᵢⱼ × I_n. Similarly, I_m ⊗ B is a block diagonal matrix with B repeated along the diagonal.

Yes, the vectorization identity vec(AXB) = (Bᵀ ⊗ A)vec(X) converts matrix equations to standard linear systems. This is fundamental in solving Sylvester and Lyapunov equations.

For A = [[a,b],[c,d]] and B = [[e,f],[g,h]], A ⊗ B = [[ae,af,bg,bh],[ag,ah,be,bf],[ce,cf,dg,dh],[cg,ch,de,df]]. This explicit formula is useful for manual computation.

In quantum computing, the Kronecker product combines individual qubit states into multi-qubit systems. The state space of n qubits is the tensor product of n 2-dimensional spaces, giving 2ⁿ dimensions.

The Tensor Product Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Tensor Product Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.