Tensor Product Calculator - 2x2 Kronecker Product Result

A tensor product calculator builds the Kronecker product A⊗B of two matrices by replacing every cell of A with that cell multiplied by the entire right matrix B. Enter the four cells of a 2x2 A and the four cells of a 2x2 B, and the result panel shows the 4x4 block matrix of four 2x2 blocks, where each block is a scalar from A times the full B.

• • Linear algebra homework: Compute the Kronecker product of two 2x2 matrices and check each 2x2 block against the a_ij * B rule.
• • Quantum mechanics states: Build the joint state vector of a two-qubit system by taking the Kronecker product of two single-qubit states.
• • Block matrix research: Test properties like non-commutativity and the block-diagonal structure of the identity Kronecker product.

The tensor product is also called the Kronecker product in matrix contexts, where the row and column counts multiply instead of needing a matching inner dimension as in ordinary matrix multiplication.

For the ordinary row-by-column product on the same two 2x2 inputs, the matrix multiplication calculator uses the dot product rule and shows both AB and BA on the same screen.

The calculator reads the eight input cells, builds 2x2 matrix A and 2x2 matrix B, and returns the 4x4 Kronecker product A⊗B. Each entry of A scales the full B to form one 2x2 block, and the four blocks are arranged in the 2x2 layout of A.

• A[i][j]: Entry in row i, column j of A. Each such entry multiplies the entire B to form one 2x2 block of the result.
• B: The right input matrix, copied as a 2x2 block into the result once for each entry of A, scaled by a different scalar each time.
• A[i][j] * B: The 2x2 block placed in the (i,j) position of the 4x4 result. Every entry of B is multiplied by the scalar A[i][j].
• (m*p) x (n*q): The shape of the result. With m = n = p = q = 2, the result is 4x4 because the rows and columns of both inputs are multiplied together.

The same block rule extends to rectangular inputs: A is m x n and B is p x q gives an (m*p) x (n*q) block matrix with m*n blocks of size p x q.

According to Wikipedia, the Kronecker product of an m x n matrix A and a p x q matrix B is the (m*p) x (n*q) block matrix whose (i,j) block is the scalar a_{ij} times B.

When the four 2x2 blocks of A⊗B are read off, the matrix transpose calculator shows the same cell-by-cell rule for the simpler case of swapping rows and columns of a single 2x2 matrix.

Four short ideas cover the block rule, the result shape, and the algebraic properties that distinguish the Kronecker product from ordinary matrix multiplication.

The Kronecker block is a scalar times a small matrix rather than a sum of products, so the dot product calculator covers the related sum-of-products rule for 2D and 3D vectors.

Enter the four cells of A and the four cells of B in row order, then read the sixteen cells of the 4x4 Kronecker product in the result panel. The output updates as you type, with each cell labelled by the scalar from A and the matching entry of B.

1. 1 Type the first row of A: Fill in the A row 1, col 1 and A row 1, col 2 cells with the top row of your first matrix. These two scalars scale the top two 2x2 blocks of the result.
2. 2 Type the second row of A: Fill in the A row 2, col 1 and A row 2, col 2 cells with the bottom row of A. These two scalars scale the bottom two 2x2 blocks of the result.
3. 3 Type the first row of B: Fill in the B row 1, col 1 and B row 1, col 2 cells with the top row of the second matrix.
4. 4 Type the second row of B: Fill in the B row 2, col 1 and B row 2, col 2 cells with the bottom row of matrix B.
5. 5 Read the top two blocks of A⊗B: The first four cells of the result panel are the top-left block (a11*B) and the top-right block (a12*B), placed side by side as a 2x4 strip.
6. 6 Read the bottom two blocks of A⊗B: The last twelve cells of the result panel are the bottom-left block (a21*B) and the bottom-right block (a22*B), placed below the top two blocks.

Suppose you are building a separable 2D image filter from a vertical 2x2 kernel A = [[1, 2], [3, 4]] and a horizontal 2x2 kernel B = [[5, 6], [7, 8]]. The result panel returns the 4x4 Kronecker product with four 2x2 blocks 1*B, 2*B, 3*B, 4*B.

If you need the element-wise sum of A and B on the same inputs, the matrix addition calculator walks through the same cell-by-cell rule on the four 2x2 entries.

The Kronecker product is built from one simple rule, but the 4x4 result is large enough that small arithmetic slips are easy to make.

• • Block-by-block result without recomputing: Each of the four 2x2 blocks of the result is a scalar from A times the full B, so the result panel writes the 4x4 matrix directly without expanding each cell by hand.
• • Visible block structure mirrors A: The result panel shows the four 2x2 blocks in the same 2x2 layout as A, so the (i,j) block can be read off the matching entry of A and B at the same time.
• • Matches the standard Kronecker product definition: The (i,j) block of A⊗B is a_{ij} times B, the same rule used in linear algebra textbooks and reference works.
• • Works for positive, negative, and decimal entries: Cells can be 0.25, 1.5, -2, or any other real number; the calculation keeps full double precision internally.
• • Sanity check before composing with other operations: Use it as a stepping stone to confirm the Kronecker product of two 2x2 matrices before plugging the result into a larger state-vector workflow.

When the Kronecker product result needs to be inverted as part of a block-matrix workflow, the matrix inverse calculator applies the cofactor and row-reduction rules on the same 2x2 input shape.

A handful of input choices and structural facts decide whether the 4x4 result represents the Kronecker product of A and B.

• • The calculator is limited to 2x2 by 2x2 Kronecker products. For 3x3 by 3x3 or m x n by p x q pairs, the same block rule applies but a larger-shape tool is needed.
• • Output values are rounded to four decimal places for display, so a true fraction like 1/3 will appear as 0.3333.

According to Wolfram MathWorld, the Kronecker product of two matrices A and B is the block matrix formed by replacing each entry a_{ij} of A with the block a_{ij} B.