Enter matrices
Use commas or spaces between values. Use a new line for each row.
Example data table
| Example | Matrix A | Matrix B | Result size | Sample use |
|---|---|---|---|---|
| Worked Example 1 | [1 2; 3 4] | [0 5; 6 7] | 4 × 4 | Learning block expansion and output growth |
| Worked Example 2 | [2 -1] | [3 4; 5 6] | 2 × 4 | Quick row-wise block scaling check |
| Worked Example 3 | [1 0; 0 1] | [7 8; 9 10] | 4 × 4 | Identity-driven repeated block placement |
Worked output for Example 1
Formula used
Definition: If A is an m × n matrix and B is a p × q matrix, then the Kronecker product is:
A ⊗ B = [ aijB ]
This means every entry aij in A multiplies the entire matrix B.
Output size: (m × p) × (n × q)
Element mapping: C[(i−1)p+r, (j−1)q+s] = aij × brs using 1-based indexing.
The calculator first validates both matrices, optionally applies transposition, then expands each value of Matrix A into a scaled copy of Matrix B.
The heatmap is drawn from the final matrix values, helping you inspect repeated blocks, zeros, signs, and magnitude clusters.
How to use this calculator
- Paste Matrix A into the first box. Use one row per line.
- Paste Matrix B into the second box using the same row format.
- Choose decimal precision and optional labels for clearer output.
- Turn on transposition when you want the calculator to use Aᵀ or Bᵀ before expansion.
- Press the calculate button to show the result above the form.
- Review the result matrix, summary metrics, and Plotly heatmap.
- Use the CSV and PDF buttons to export the current output.
Frequently asked questions
1) What does the Kronecker product do?
It builds a larger block matrix by multiplying every entry of the first matrix by the entire second matrix. The result preserves the structure of the first matrix while repeating scaled copies of the second.
2) Is the Kronecker product the same as regular matrix multiplication?
No. Regular matrix multiplication combines rows and columns through dot products. The Kronecker product expands matrix size by replacing each entry of the first matrix with a scaled block formed from the second.
3) How large will the result matrix be?
If Matrix A is m × n and Matrix B is p × q, the result becomes (m×p) × (n×q). Even small inputs can produce very large outputs, so always check dimensions first.
4) Why would I transpose a matrix before calculation?
Transposition changes row and column placement before the block expansion occurs. This is useful when matching textbook conventions, verifying identities, or preparing inputs for vectorization, tensor, and systems applications.
5) What input separators are supported?
You can separate values with spaces or commas. Each row should be on a new line. The calculator also treats semicolons as row breaks, which is convenient for compact matrix entry.
6) What does the heatmap show?
The Plotly heatmap visualizes the final matrix values. It helps you spot repeated blocks, zero regions, sign changes, and areas with strong magnitude concentration that may be harder to detect in a large table.
7) Can I export the result for reports or class notes?
Yes. The CSV button downloads the result matrix as a spreadsheet-friendly file. The PDF button exports a summary and matrix table, which is useful for reports, assignments, revision sheets, and documentation.
8) Where is the Kronecker product commonly used?
It appears in linear algebra, tensor analysis, signal processing, quantum computing, image models, control systems, and structured matrix equations. It is especially useful when separable patterns or repeated block relationships are present.