How this calculator works
The tensor product of two matrices, also called the Kronecker product, builds a block matrix. Each entry aij of A is multiplied by the entire matrix B, and the blocks are arranged in the same pattern as the entries of A. The result has the product of the row counts and the product of the column counts. The operation is bilinear and appears in signal processing, quantum mechanics and systems of equations with repeated structure.
Formula
(A tensor B) is the block matrix whose (i,k) block is aij B. If A is m by n and B is p by q, the result is mp by nq.
Worked example
Taking A with rows 1, 2 and 3, 4 and B with rows 0, 5 and 6, 7 gives a 4 by 4 block matrix. The first block is 1 times B, namely 0, 5; 6, 7, and the second block is 2 times B, namely 0, 10; 12, 14. The full product is 0, 5, 0, 10; 6, 7, 12, 14; 0, 15, 0, 20; 18, 21, 24, 28.
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Frequently asked questions
Is the tensor product the same as the Hadamard product?
No. The Hadamard product multiplies matching entries of two matrices of the same shape. The tensor product builds a larger block matrix from every entry of one matrix and the whole of the other.
Does the order matter?
Yes. A tensor B and B tensor A generally have different shapes and entries, although they are related by a permutation of rows and columns.
What is the determinant of a Kronecker product?
For square A of size m and square B of size n, det(A tensor B) = det(A)^n det(B)^m. This follows from the eigenvalue products of each factor.