1. Definition & formula
The Kronecker product (also called tensor product, denoted $\otimes$) of an $m\times n$ matrix $A=[a_{ij}]$ and a $p\times q$ matrix $B$ is the block matrix:
The result has size $(mp)\times(nq)$. Each entry $a_{ij}$ of $A$ multiplies the entire matrix $B$ to form the corresponding block.
2. Useful properties
- Mixed-product property: $(A\otimes B)(C\otimes D) = (AC)\otimes(BD)$ when dimensions are compatible.
- Transpose: $(A\otimes B)^{T} = A^{T}\otimes B^{T}$.
- Associativity: $(A\otimes B)\otimes C = A\otimes(B\otimes C)$ (up to natural reshaping).
- Scalar multiple: $(\alpha A)\otimes B = A\otimes(\alpha B) = \alpha (A\otimes B)$.
- Eigenvalues: If $A v = \lambda v$ and $B w = \mu w$, then $(A\otimes B)(v\otimes w) = (\lambda\mu)(v\otimes w)$.
3. Worked examples
Example 1 — small matrices
Compute $A\otimes B$ by replacing each $a_{ij}$ with $a_{ij}B$:
Result is $(2\cdot2)\times(2\cdot2)=4\times4$.
Example 2 — with vector
$B$ is $1\times2$, $A\otimes B$ is $(2\cdot1)\times(2\cdot2)=2\times4$:
4. Common mistakes
- Confusing element-wise multiply with Kronecker product: Kronecker creates large block matrices, not entrywise multiplication.
- Expecting same size: $A\otimes B$ rarely has the same dimension as $A$ or $B$ unless one is $1\times1$.
- Forgetting block order: Blocks follow row-major order of $A$ — each row of $A$ creates a block row of $B$-blocks.
- Mismatching dimensions when using mixed-product property: ensure matrix multiplications $AC$ and $BD$ are valid before applying the property.
- Indexing mistakes: when programming, carefully map $(i,j)$ in $A$ to blocks of size $p\times q$ in the result.
5. Practice problems
Try these Kronecker product problems. Click Show Answer to reveal the result.
Exercise 1
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Exercise 2
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Exercise 3
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Here $A$ is $1\times3$, $B$ is $2\times1$, so $A\otimes B$ is $(1\cdot2)\times(3\cdot1)=2\times3$:
$$ A\otimes B = \begin{pmatrix} 1\cdot B & 2\cdot B & 3\cdot B \end{pmatrix} = \begin{pmatrix} 4 & 8 & 12 \\ 5 & 10 & 15 \end{pmatrix}. $$Exercise 4
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Carefully multiply scalar blocks; keep block positions aligned with $A$'s entries.
The Kronecker product also appears with eigenvalues, SVD, and QR decomposition.
