Enter Your Matrices
Use commas or spaces between numbers. Use new lines or semicolons between rows.
Formula Used
If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is an mp × nq block matrix.
Useful identities included in this calculator:
- size(A ⊗ B) = (m p) × (n q)
- rank(A ⊗ B) = rank(A) × rank(B)
- trace(A ⊗ B) = trace(A) × trace(B), when both are square
- det(A ⊗ B) = det(A)p × det(B)m, when A and B are square
- ||A ⊗ B||F = ||A||F × ||B||F
How to Use This Calculator
- Enter Matrix A in the first box using commas or spaces between entries.
- Enter Matrix B in the second box using the same format.
- Separate rows with new lines or semicolons.
- Choose the decimal precision you want for displayed values.
- Press Calculate Kronecker Product to generate the output matrix.
- Review the summary cards for size, rank, norm, trace, determinant, and matrix statistics.
- Inspect the heatmap to study block structure and entry magnitude.
- Download the result as CSV or PDF when needed.
Example Data Table
| Example Item | Value |
|---|---|
| Matrix A | [1 2; 3 4] |
| Matrix B | [0 5; 6 7] |
| Output Size | 4 × 4 |
| Output Row 1 | [0, 5, 0, 10] |
| Output Row 2 | [6, 7, 12, 14] |
| Output Row 3 | [0, 15, 0, 20] |
| Output Row 4 | [18, 21, 24, 28] |
FAQs
1. What does the Kronecker product represent?
It builds a larger block matrix by multiplying every entry of Matrix A by the entire Matrix B. This operation is common in linear algebra, signal processing, tensor models, quantum systems, and structured matrix design.
2. How should I type my matrices?
Enter numbers with commas or spaces between columns. Use new lines or semicolons to separate rows. For example, type 1, 2 on one row and 3, 4 on the next row.
3. What size will the output matrix have?
If Matrix A is m × n and Matrix B is p × q, the Kronecker product has size mp × nq. The calculator shows this immediately in the summary section above the form.
4. Does the calculator support decimals and negative values?
Yes. You can enter integers, decimals, and negative numbers. The precision field controls how many decimal places are displayed in the matrix output, summary cards, CSV export, and PDF export.
5. Why are rank, trace, and determinant shown?
These properties help verify matrix identities and support deeper analysis. Rank shows linear independence, trace summarizes diagonal behavior, and determinant gives scaling information for square matrices under the Kronecker product rules.
6. What does the heatmap show?
The heatmap visualizes the numeric structure of the resulting matrix. It helps you spot repeated blocks, zero patterns, sign changes, and relative magnitude differences across the full Kronecker product.
7. When is the determinant relation available?
It appears when both input matrices are square. In that case, the calculator applies the identity det(A ⊗ B) = det(A)p × det(B)m, where p is B’s order and m is A’s order.
8. Can I export the result for reports or homework?
Yes. Use the CSV button for spreadsheet-friendly data and the PDF button for a printable summary. Both export options use the current matrix values and displayed table content.