Tensor Product Calculator

The tensor product of matrices (also called the Kronecker product) is a binary operation that takes two matrices A (of size rA×cA) and B (of size rB×cB) and produces a block matrix of size (rA·rB)×(cA·cB). Each element aᵢⱼ of matrix A is multiplied by the entire matrix B to form a sub-block in the result. It is widely used in quantum computing, signal processing, and linear algebra.

To compute A⊗B, replace each element aᵢⱼ of matrix A with the scaled matrix aᵢⱼ·B. Arrange these scaled sub-matrices in the same grid pattern as the original elements of A. For example, for a 2×2 matrix A and 2×2 matrix B, the result is a 4×4 block matrix composed of four 2×2 sub-blocks.

The formula is: (A⊗B)[(i-1)·rB+k, (j-1)·cB+l] = aᵢⱼ · bₖₗ, where i,j index into A and k,l index into B. The resulting matrix has dimensions (rA·rB) × (cA·cB).

If matrix A has rA rows and cA columns, and matrix B has rB rows and cB columns, then the Kronecker product A⊗B will have (rA × rB) rows and (cA × cB) columns. For example, a 2×3 matrix tensor-producted with a 4×2 matrix yields an 8×6 result matrix.

No, the Kronecker product is generally not commutative. A⊗B and B⊗A are not equal in general, though they are related by permutation matrices — they are permutation-equivalent. The sizes of A⊗B and B⊗A are the same only if A and B have the same dimensions.

Yes, the Kronecker product is associative. This means (A⊗B)⊗C = A⊗(B⊗C). This property makes it straightforward to extend the operation to chains of three or more matrices without worrying about the order of grouping.

In the context of matrices, yes — the tensor product and Kronecker product refer to the same operation. In abstract mathematics, the tensor product is a more general concept that applies to vector spaces and linear maps. The Kronecker product is the concrete matrix representation of the tensor product when applied to matrices with a chosen basis.

The Kronecker product appears in quantum computing (representing multi-qubit states and gates), image processing, solving linear matrix equations like the Sylvester equation, graph theory (Kronecker graphs), and constructing large structured matrices from smaller ones in numerical methods.