Free Tensor Product Calculator

What Is the Kronecker Product?

The Kronecker product (also referred to as the matrix tensor product or matrix direct product) is a specialized binary operation that combines two matrices of arbitrary dimensions. Unlike ordinary matrix multiplication, which requires matching inner dimensions, the Kronecker product works on any pair of matrices and produces a new matrix whose size is the product of the original dimensions. Precisely, if matrix $A$ has $r_A$ rows and $c_A$ columns, and matrix $B$ has $r_B$ rows and $c_B$ columns, their Kronecker product $A \otimes B$ will have $r_A \cdot r_B$ rows and $c_A \cdot c_B$ columns. When one of the inputs is a vector (single row or column), the operation is often called the vector tensor product.

This free online tensor product calculator — often called a Kronecker product calculator — automates the computation, letting you focus on understanding the structure and properties of the result. Whether you need the Kronecker product of $2\times2$ matrices or larger rectangular matrices, the tool handles it instantly.

How to Compute the Tensor Product

There are two common ways to calculate the Kronecker product by hand:

1. Block‑matrix method – Multiply each element of $A$ by the whole matrix $B$ and arrange the resulting blocks:

$$A \otimes B = \begin{bmatrix} a_{11}B & a_{12}B & \cdots & a_{1c_A}B\\ a_{21}B & a_{22}B & \cdots & a_{2c_A}B\\ \vdots & \vdots & \ddots & \vdots\\ a_{r_A1}B & a_{r_A2}B & \cdots & a_{r_Ac_A}B \end{bmatrix}$$

Here $a_{ij}B$ means every entry of $B$ is multiplied by the scalar $a_{ij}$.

2. Element‑wise formula – For a given index pair $(i,j)$ in the result, let $r_B$ and $c_B$ be the row/column counts of $B$. Then

$$(A \otimes B)_{ij} = A_{\lceil i / c_B \rceil,\; \lceil j / c_B \rceil} \;\cdot\; B_{\, (i-1)\bmod r_B + 1,\; (j-1)\bmod c_B + 1}.$$

Both approaches are equivalent; the block method is usually easier for small matrices, while the formula is handy for large or general computations.

Kronecker Product of $2\times2$ Matrices

Take

$$A = \begin{pmatrix}a&b\\c&d\end{pmatrix},\qquad B = \begin{pmatrix}e&f\\g&h\end{pmatrix}.$$

Using the block method:

$$A\otimes B = \begin{pmatrix} aB & bB\\ cB & dB \end{pmatrix} = \begin{pmatrix} ae & af & be & bf\\ ag & ah & bg & bh\\ ce & cf & de & df\\ cg & ch & dg & dh \end{pmatrix}.$$

Thus the Kronecker product of two $2\times2$ matrices yields a $4\times4$ result. This example can be verified immediately with the tensor product calculator.

Properties of the Kronecker Product

The Kronecker product satisfies several algebraic properties that are useful both in theory and in applications.

• Associativity: $(A \otimes B) \otimes C = A \otimes (B \otimes C)$ for any compatible matrices.
• Bilinearity: The operation is linear in each argument separately:

$$(A+B)\otimes C = A\otimes C + B\otimes C,\quad A\otimes (B+C) = A\otimes B + A\otimes C,$$

and for a scalar $x$, $(xA)\otimes B = x(A\otimes B) = A\otimes (xB)$.

• Transposition: $(A\otimes B)^T = A^T \otimes B^T$. The same holds for conjugate transpose: $(A\otimes B)^* = A^* \otimes B^*$.

• Singular values: If the non‑zero singular values of $A$ are $\sigma_1,\dots,\sigma_{p_A}$ and those of $B$ are $s_1,\dots,s_{p_B}$, then the non‑zero singular values of $A\otimes B$ are all products $\sigma_i s_j$. Consequently, $\operatorname{rank}(A\otimes B) = \operatorname{rank}(A) \cdot \operatorname{rank}(B)$.

• Invertibility: If $A$ and $B$ are square and invertible, then $A\otimes B$ is also invertible and $(A\otimes B)^{-1} = A^{-1} \otimes B^{-1}$.

• Eigenvalues (for square $A$ of size $m$ and square $B$ of size $n$): Let $\alpha_1,\dots,\alpha_m$ be the eigenvalues of $A$ (with multiplicities) and $\beta_1,\dots,\beta_n$ those of $B$. Then the eigenvalues of $A\otimes B$ are $\alpha_i\beta_j$ for $i=1,\dots,m$ and $j=1,\dots,n$.

From the eigenvalue property two useful formulas follow directly:

$$\det(A\otimes B) = (\det A)^n \, (\det B)^m, \qquad \operatorname{tr}(A\otimes B) = (\operatorname{tr} A)\,(\operatorname{tr} B).$$

Using the Tensor Product Calculator

To compute the Kronecker product with this tool, first select the dimensions of your matrices (number of rows and columns for each). Then enter the numeric entries in the displayed fields. The calculator will instantly display the resulting block matrix. Keep in mind that the Kronecker product is not commutative: $A\otimes B$ generally differs from $B\otimes A$, so the order you enter the matrices matters.