Calculator Inputs
Use vectors for outer products or matrices for Kronecker-style tensor products.
Example Data Table
| Mode | First Operand | Second Operand | Expected Output Size | Why It Matters |
|---|---|---|---|---|
| Vector ⊗ Vector | [1, 2, 3] | [4, 5] | 3 × 2 | Shows an outer product with rank-one structure. |
| Matrix ⊗ Matrix | [[1, 2], [3, 4]] | [[0, 5], [6, 7]] | 4 × 4 | Illustrates block scaling and repeated local patterns. |
| Matrix ⊗ Matrix | [[2, 0], [1, -1]] | [[1, 3, 2]] | 2 × 6 | Useful for rectangular operands and dimension checks. |
Formula Used
Vector tensor product: If u = [u1, ..., um] and v = [v1, ..., vn], then
(u ⊗ v)ij = uivj
Matrix tensor product: If A is m × n and B is p × q, then A ⊗ B is an (mp) × (nq) matrix, where
(A ⊗ B)(i-1)p+r,\,(j-1)q+s = aijbrs
This means every entry of the first operand scales the entire second operand, producing a larger structured matrix composed of repeated blocks.
How to Use This Calculator
- Choose Vector ⊗ Vector for outer products or Matrix ⊗ Matrix for block tensor products.
- Enter numeric values into both operand boxes. Use commas or spaces between values. In matrix mode, separate rows with semicolons or line breaks.
- Set the output precision to match your coursework, reporting style, or verification needs.
- Press Compute Tensor Product to display the result above the form.
- Review the result matrix, quick statistics, and the heatmap pattern.
- Use the CSV or PDF buttons to export your result for notes, reports, or classroom work.
Frequently Asked Questions
1. What does this calculator compute?
It computes tensor products for two vectors or two matrices. In vector mode, it returns an outer product. In matrix mode, it returns the Kronecker-style block expansion commonly used in algebra, quantum mechanics, and linear systems.
2. What is the difference between outer product and tensor product here?
For vectors, the calculator uses the standard outer product, which is a concrete matrix form of the tensor product. For matrices, it uses the Kronecker construction, which is the familiar computational representation of a tensor product.
3. How should I type matrices?
Enter one row at a time. Separate values with spaces or commas. Separate rows with semicolons or line breaks. For example: 1 2; 3 4 or two lines containing 1 2 and 3 4.
4. What size will the output have?
If the first matrix is m × n and the second is p × q, the output is (mp) × (nq). For vectors of lengths m and n, the output becomes an m × n matrix.
5. Why is the heatmap useful?
The heatmap makes repeated block structure easier to see. It quickly reveals scaling patterns, zero regions, and symmetry-like behavior that may be less obvious in a raw numeric table.
6. Can I export my result?
Yes. After computation, the page provides CSV and PDF export buttons. CSV is useful for spreadsheets and further processing. PDF is useful for homework, notes, and printable documentation.
7. Why do I get an input error?
Input errors usually happen when one value is not numeric or a matrix row has a different number of entries than the others. Check separators, remove stray symbols, and make sure each row is consistent.
8. Where are tensor products used?
Tensor products appear in linear algebra, quantum computing, signal processing, image transforms, systems theory, and multilinear modeling. They help combine spaces, operators, and structured data into larger composite forms.