Tensor Product Calculator
Calculate the tensor product (Kronecker product) of two matrices.
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Matrix A
Matrix B
Tensor Product (Kronecker Product)
The tensor product of two matrices A and B is defined as:
\[ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B & \cdots \\ a_{21}B & a_{22}B & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix} \]
Properties:
- \((A \otimes B)(C \otimes D) = (AC) \otimes (BD)\)
- \((A \otimes B)^T = A^T \otimes B^T\)
- \((A \otimes B)^{-1} = A^{-1} \otimes B^{-1}\)
- \(\text{det}(A \otimes B) = \text{det}(A)^n \text{det}(B)^m\)
- \(\text{tr}(A \otimes B) = \text{tr}(A)\text{tr}(B)\)
Applications:
- Quantum mechanics
- Signal processing
- Image processing
- Neural networks
- Multilinear algebra
Geometric Interpretation:
- Represents the outer product of vector spaces
- Combines two linear transformations
- Preserves tensor structure
- Maintains multilinearity