Tensor Product Calculator

The tensor product (specifically the Kronecker product) takes an m-by-n matrix A and a p-by-q matrix B, resulting in a large block matrix of dimension (m times p) by (n times q). If matrix A has elements a_ij, every individual element inside matrix A is multiplied by the entire matrix B. Mathematically, the resulting block matrix C is structured as a_ij multiplied by B for all valid indices. The final dimensions of the output matrix scale multiplicatively based on the input row and column sizes.

Let us compute the tensor product of a 2x1 matrix A and a 2x1 matrix B. Assume matrix A has elements [a1 = 2, a2 = 3] and matrix B has elements [x1 = 4, x2 = 5]. Following the underlying equations (such as p1 = a1 * x1 and kk4 = a2 * x2), each element of matrix A multiplies each element of matrix B. For element a1 (which is 2), we multiply it by vector B [4, 5], yielding [8, 10]. For element a2 (which is 3), we multiply it by vector B [4, 5], yielding [12, 15]. The combined resultant tensor product matrix yields the column vector [8, 10, 12, 15]^T.

When working with tensor products, always double-check the dimensions of your input matrices; unlike standard matrix multiplication, the Kronecker product does not require matching inner dimensions. Keep track of your indexing conventions (row-major vs column-major) to ensure your output block matrices are assembled in the correct quadrants. Finally, remember that intermediate scaling factors scale linearly with the product of the component matrix dimensions.