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Kronecker Product Calculator

Enter the entries of matrices A and B to compute their Kronecker product.

How it is calculated

The Kronecker product A ⊗ B replaces every entry of A by that entry times the whole of B. For 2×2 matrices, block (i, j) of the 4×4 result is A[i][j] · B, so the entry in row 2i + p and column 2j + q is A[i][j] × B[p][q] (i, j, p, q each 0 or 1). Enter the eight numbers a11…a22 and b11…b22; any real numbers are accepted, including zero and negatives, and a blank or invalid field gives no result. The output is the 4×4 matrix, each entry shown with two decimals. The order matters: A ⊗ B is generally not equal to B ⊗ A, although the two have the same entries rearranged. The tool covers 2×2 matrices only.

Worked example

Let A = [[1, 2], [3, 4]] and B = [[0, 5], [6, 7]]. Step 1: the top-left 2×2 block is 1 × B = [[0, 5], [6, 7]]. Step 2: the top-right block is 2 × B = [[0, 10], [12, 14]]. Step 3: the bottom-left block is 3 × B = [[0, 15], [18, 21]] and the bottom-right is 4 × B = [[0, 20], [24, 28]]. Step 4: putting the blocks together gives the first row 0, 5, 0, 10 and the last row 18, 21, 24, 28, as in the 4×4 result the tool prints, with entries shown to two decimals.

Try this example →

Common mistakes

  • Confusing the Kronecker product with ordinary matrix multiplication. The result is 4×4 here, not 2×2, and the entries are products of single entries, not sums of row-by-column products.
  • Swapping the order. A ⊗ B puts copies of B inside A's pattern; B ⊗ A puts copies of A inside B's pattern, and the two matrices differ.
  • Mixing up the block position. The entry A[i][j] scales the whole block in block-row i and block-column j, so a12 sits in the top-right block, not in the second column of the first row.
  • Typing a matrix by columns instead of by rows. a12 is row 1, column 2, and a21 is row 2, column 1.
  • Expecting the tool to handle larger matrices. It is for two 2×2 matrices; for 3×3 or other sizes the block rule is the same but you must apply it by hand.

About this calculator

The Kronecker product A ⊗ B of two matrices is a block matrix formed by multiplying every entry of A by the entire matrix B, so two 2×2 matrices produce a 4×4 result — unlike ordinary matrix multiplication, it works on matrices of any dimensions and combines them into a much larger structured matrix rather than reducing them to a single product of compatible size.

Because the Kronecker product describes how independent systems combine, it's central to quantum computing (the joint state of two qubits is the tensor product of their individual states), control theory and systems engineering (modeling combined linear systems), and signal processing. Quantum computing researchers, control systems engineers, and linear algebra students use it to construct the state space or system matrix of a combined system from its individual components.

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Comments (1)

  • Editorial team

    Order matters here: A ⊗ B and B ⊗ A usually give different 4x4 matrices, though they contain the same entries rearranged. Swap the two inputs to see it. A check for 2x2 inputs is that the determinant of the result equals det(A) squared times det(B) squared. Results are displayed with two decimals. Which application are you building these for?

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