Tensor Product Calculator

About Tensor Product Calculator

The Tensor Product Calculator computes the matrix tensor product A ⊗ B, also known as the Kronecker product. It accepts rectangular matrices, preserves exact rational arithmetic when possible, and visualizes the defining block structure: every entry of Matrix A expands into a full scaled copy of Matrix B.

Tensor Product Formula

If A is an m × n matrix and B is a p × q matrix, then A ⊗ B is an mp × nq matrix. The block form is:

Equivalently, each entry is indexed by:

How to Use This Calculator

• Enter Matrix A with one row per line, using spaces or commas between entries.
• Enter Matrix B in the same format. Matrix A and Matrix B may both be rectangular.
• Choose exact fraction output for symbolic work, or decimal output for compact numerical results.
• Click Compute Tensor Product to see the result matrix, dimensions, block expansion, and copyable formats.

Tensor Product vs Matrix Multiplication

Important Properties

Bilinearity

The tensor product distributes over matrix addition and scalar multiplication: (A + C) ⊗ B = A ⊗ B + C ⊗ B and (kA) ⊗ B = k(A ⊗ B).

Mixed Product Property

When the ordinary products are defined, the Kronecker product satisfies:

This identity is one reason tensor products are useful for structured linear systems and separable operators.

Transpose and Inverse

The transpose follows (A ⊗ B)^T = A^T ⊗ B^T. If both square matrices are invertible, then (A ⊗ B)⁻¹ = A⁻¹ ⊗ B⁻¹.

Where Tensor Products Are Used

• Quantum computing: multi-qubit gates and combined quantum states are represented with Kronecker products.
• Signal and image processing: separable filters and two-dimensional transforms often use tensor product structure.
• Numerical linear algebra: large structured matrices can be stored or applied efficiently using Kronecker factors.
• Graph theory: graph product adjacency matrices are often expressed through Kronecker-style operations.
• Statistics and machine learning: covariance structures, Gaussian processes, and multidimensional grids can use tensor product matrices.

FAQ

What is the tensor product of two matrices?

For matrices A of size m by n and B of size p by q, the tensor product A ⊗ B is the mp by nq block matrix formed by replacing each entry a_ij of A with the scaled block a_ijB.

Is the tensor product the same as the Kronecker product?

For finite matrices, the terms tensor product and Kronecker product are commonly used for the same block matrix operation. The notation A ⊗ B is standard in linear algebra, quantum computing, signal processing, and numerical methods.

What size is A ⊗ B?

If A has m rows and n columns, and B has p rows and q columns, then A ⊗ B has mp rows and nq columns. Every row of A expands into p rows, and every column of A expands into q columns.

Does the order matter in A ⊗ B?

Yes. In general A ⊗ B is not the same matrix as B ⊗ A, even though the two products contain related scaled blocks. The ordering controls how the row and column indices are arranged.

Can this calculator use fractions?

Yes. Entries such as 1/2, -3/4, 0.25, and 2e-3 are accepted. Exact fraction mode keeps rational values exact throughout the tensor product.