Tensor Product Calculator
Compute a Kronecker product with MiniWebtool’s Tensor Product Calculator. Two rectangular matrices produce an exact-fraction result, with block-by-block visualization and copyable output explaining how each entry expands the second matrix.
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Enter two matrices below to get A ⊗ B, shown block by block with exact fractions.
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.
At a Glance
Tensor Product Calculator computes the Kronecker product A ⊗ B of two matrices. It shows the result in exact fractions or decimals, along with the block structure and copyable output.
| Feature | What you get | Good for |
|---|---|---|
| Matrix input | Enter each matrix with one row per line and spaces or commas between entries. Both matrices may be rectangular. | Matrices of different sizes with no matching inner dimension |
| Number formats | Accepts integers, decimals and fractions such as 1/2, -3/4, 0.25 and 2e-3. | Mixing fractions and decimals in one matrix |
| Output precision | Choose exact fractions, decimals with 6 significant digits, or decimals with 10 significant digits. | Symbolic work or compact numerical results |
| Result size | For A of m × n and B of p × q, the result is mp × nq. A dimension map shows the sizes involved. | Checking dimensions before using the product in another step |
| Block expansion | Shows how each entry of A scales a full copy of B, with block boundaries highlighted and one block expanded. | Learning how the Kronecker product is built |
| Result statistics | Reports dimensions, sparsity, Frobenius norm, minimum entry and maximum entry. | Quick sanity checks on a large result |
| Copy formats | Copy the result as a space-separated matrix, CSV or a LaTeX bmatrix. | Pasting into a spreadsheet, script or paper |
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
| 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:
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
- 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 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
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