Tensor (Kronecker) Product Calculator

Compute the Kronecker product of two 2x2 matrices, producing a 4x4 result matrix.

Enter Matrices

Matrix A (2x2)
Matrix B (2x2)

Result: A ⊗ B (4x4)

Result dimensions4 x 4
Number of elements16
Sum of all elements--
Trace (sum of diagonal)--

Step-by-Step Solution

(A ⊗ B)[i,j] = A[ceil(i/q), ceil(j/q)] * B[i mod q, j mod q]

Understanding the Tensor (Kronecker) Product

The Kronecker product (also called the tensor product of matrices) is an operation on two matrices of arbitrary size resulting in a block matrix. For an m-by-n matrix A and a p-by-q matrix B, the Kronecker product A ⊗ B is an mp-by-nq matrix formed by replacing each element aij of A with the block aij * B.

Key Properties

Block Structure

Each entry of A is replaced by that entry multiplied by the entire matrix B.

A ⊗ B = [a_ij * B]

Not Commutative

In general, A ⊗ B is not equal to B ⊗ A (order matters).

A ⊗ B != B ⊗ A

Associative

The Kronecker product is associative: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C).

(A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)

Mixed-Product Property

(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD) when the products AC and BD exist.

(A⊗B)(C⊗D) = AC ⊗ BD

Applications

  • Quantum computing: constructing multi-qubit state spaces and gates
  • Signal processing: multi-dimensional filtering
  • Statistics: covariance structures for multivariate data
  • Control theory: modeling interconnected systems
  • Computer graphics: texture synthesis