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Tensor Product Calculator

Compute the tensor product, also called the Kronecker product, of two rectangular matrices with exact fraction arithmetic, block-by-block visualization, copyable results, and SEO-friendly linear algebra explanations.

Tensor Product Calculator
Quick examples:
2×2 block expansion Fraction entries Identity × rectangular Quantum Pauli-style
Put each row on a new line. Separate entries with spaces or commas. Fractions like 1/3 and decimals like 0.25 are accepted.
The result has rows(A) × rows(B) rows and cols(A) × cols(B) columns.

Each entry of A becomes a full copy of B, scaled by that entry. Hover matching blocks in the result to see the expansion map.

Embed Tensor Product Calculator Widget

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:

A ⊗ B = [ a11B a12B ... a1nB a21B a22B ... a2nB ... am1B am2B ... amnB ]

Equivalently, each entry is indexed by:

C[(i−1)p + r, (j−1)q + s] = A[i,j] × B[r,s]

How to Use This Calculator

Tensor Product vs Matrix Multiplication

Operation Input requirement Output size Main idea
Matrix multiplication AB columns(A) = rows(B) rows(A) × columns(B) Dot products combine rows of A with columns of B.
Tensor product A ⊗ B No inner-dimension match required rows(A)rows(B) × columns(A)columns(B) Every entry of A scales a complete copy of B.
Element-wise product A ⊙ B A and B must have the same shape same shape as A and B Corresponding entries are multiplied one by one.

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:

(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD)

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 = AT ⊗ BT. If both square matrices are invertible, then (A ⊗ B)−1 = A−1 ⊗ B−1.

Where Tensor Products Are Used

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 aij of A with the scaled block aijB.

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.

Reference this content, page, or tool as:

"Tensor Product Calculator" at https://MiniWebtool.com/tensor-product-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Apr 24, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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