Kronecker Product Calculator

Verified Calculation Engine

The online Kronecker Product Calculator helps you calculate instantly and solve problems related to Matrices Determinants. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Engineering Math calculations.

Compute the Kronecker product A ⊗ B of two matrices. If A is m×n and B is p×q, the result is the (mp)×(nq) block matrix of scaled copies of B. Dimensions of A and B need not match.

A ⊗ B = [a_ij B] with size (m p)×(n q)
The Kronecker product builds a block matrix whose (i,j)-th block is a_ij times B. Unlike matrix multiplication, A and B do not need compatible inner dimensions.
  • Matrix A is m×n and Matrix B is p×q with real entries.
  • Result size is always (m·p)×(n·q); A and B need not have matching dimensions.
  • This operation is not ordinary matrix multiplication.

Inputs

2×2
Edit cells directly. Use Rows and Columns to set the matrix size.Left factor in A ⊗ B. Any m×n size (independent of B).
Please enter a valid matrix.
2×2
Edit cells directly. Use Rows and Columns to set the matrix size.Right factor in A ⊗ B. Any p×q size (independent of A).
Please enter a valid matrix.

Results

Worked Examples
2×2 ⊗ 2×2

Classic square Kronecker product yielding a 4×4 matrix.

Inputs:
  • matrix1: [ [ 1, 2 ], [ 3, 4 ] ]
  • matrix2: [ [ 0, 5 ], [ 6, 7 ] ]
Expected Outputs:
  • result: [0,5,0,10], [6,7,12,14], [0,15,0,20], [18,21,24,28]
  • dimensions: 4×4
Rectangular 2×3 ⊗ 3×2

Non-square factors: result is 6×6 (no matching-dimension requirement).

Inputs:
  • matrix1: [ [ 1, 2, 3 ], [ 4, 5, 6 ] ]
  • matrix2: [ [ 7, 8 ], [ 9, 10 ], [ 11, 12 ] ]
Expected Outputs:
  • dimensions: 6×6
1×1 ⊗ matrix (scalar block)

[[3]] ⊗ B equals 3B.

Inputs:
  • matrix1: [ [ 3 ] ]
  • matrix2: [ [ 1, 2 ], [ 3, 4 ] ]
Expected Outputs:
  • result: [3,6], [9,12]
  • dimensions: 2×2

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About this calculator

Overview

The Kronecker Product Calculator computes A ⊗ B, the block matrix formed by scaling B by every entry of A. If A is m×n and B is p×q, A ⊗ B is (mp)×(nq). Unlike ordinary matrix multiplication, the dimensions of A and B do not need to match.

When to use

Use this tool for tensor-product constructions, Lyapunov/Sylvester operators in vectorized form, multi-particle or multi-factor linear models, and checking hand calculations of block expansions.

Inputs explained

  • Matrix A — left factor (any m×n)
  • Matrix B — right factor (any p×q)

Formula / method

For each entry a_ij of A, place the block a_ij·B into the (i,j) block position of the result. The resulting matrix has row blocks of height p and column blocks of width q. Display dimensions of A, B, and A ⊗ B.

Worked example

For A=[[1,2],[3,4]] and B=[[0,5],[6,7]], A ⊗ B is the 4×4 matrix [[0,5,0,10],[6,7,12,14],[0,15,0,20],[18,21,24,28]].

Interpreting results

Each scalar in A multiplies an entire copy of B. The Kronecker product is associative and bilinear, and satisfies (A⊗B)(C⊗D)=(AC)⊗(BD) when the ordinary products exist.

Assumptions

  • Entries are real numbers.
  • Empty matrices are rejected.
  • Floating-point rounding may affect near-integer display.

Limitations

  • Large factors produce very large results (size multiplies).
  • Not a substitute for ordinary matrix multiplication AB.
  • Educational tool — verify critical designs independently.

Important note

Educational engineering-mathematics tool. Verify critical designs with hand analysis or certified software.

How to Use This Calculator

  1. Enter the required values in the input fields.
  2. Click the Calculate button.
  3. View the computed result instantly.

Formula Used

For each entry a_ij of A, place the block a_ij·B into the (i,j) block position of the result. The resulting matrix has row blocks of height p and column blocks of width q. Display dimensions of A, B, and A ⊗ B.

Example Calculation

For A=[[1,2],[3,4]] and B=[[0,5],[6,7]], A ⊗ B is the 4×4 matrix [[0,5,0,10],[6,7,12,14],[0,15,0,20],[18,21,24,28]].

Frequently Asked Questions

What is Kronecker Product Calculator?

The Kronecker Product Calculator computes A ⊗ B, the block matrix formed by scaling B by every entry of A. If A is m×n and B is p×q, A ⊗ B is (mp)×(nq). Unlike ordinary matrix multiplication, the dimensions of A and B do not need to match.

How does Kronecker Product Calculator work?

For each entry a_ij of A, place the block a_ij·B into the (i,j) block position of the result. The resulting matrix has row blocks of height p and column blocks of width q. Display dimensions of A, B, and A ⊗ B.

Why use this Engineering Math calculator?

Use this tool for tensor-product constructions, Lyapunov/Sylvester operators in vectorized form, multi-particle or multi-factor linear models, and checking hand calculations of block expansions.