Calculadora de matrices

1. Definición y fórmula

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. Propiedades útiles

• 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. Ejemplos resueltos

Ejemplo 1: matrices pequeñas

Ejemplo 2: con vector

4. Errores comunes

• Confundir la multiplicación elemento a elemento con el producto de Kronecker: El producto de Kronecker crea grandes matrices por bloques, no una multiplicación entrada a entrada.
• Esperar el mismo tamaño: $A\otimes B$ rarely has the same dimension as $A$ or $B$ unless one is $1\times1$.
• Olvidar el orden de los bloques: 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.
• Errores de indexación: 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 Mostrar respuesta to reveal the result.

Ejercicio 1

Ejercicio 2

Ejercicio 3

Ejercicio 4