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Manuals/calci/MATRIXTENSORPRODUCT (view source)
Revision as of 19:31, 12 July 2017
, 19:31, 12 July 2017no edit summary
*Tensor product is different from general product.
*Tensor product is different from general product.
*The Tensor product is defined by the product two vector spaces V and W is itself a Vector space.
*The Tensor product is defined by the product two vector spaces V and W is itself a Vector space.
*It is denoted by VSymbol W.
*It is denoted by <math>V\otimes W</math>.
*The tensor product of V and W is the vector space generated by the symbols v\otimes w v\otimes w, with v belongs to V and w belongs to W.
*The tensor product of V and W is the vector space generated by the symbols <math>v\otimes w </math>, with <math>v \isin V</math> and <math>w \isin W</math>.
*The tensor product from the direct sum vector space, whose dimension is the sum of the dimensions of the two summands:Now consider any 2x2 matrices
*The tensor product from the direct sum vector space, whose dimension is the sum of the dimensions of the two summands:
<math>dim (V \otimes W)= dim V +dim W </math>
*Now consider any 2x2 matrices:
<math>\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}\otimes \begin{bmatrix}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{bmatrix} =
\begin{bmatrix}
a_{11}\begin{bmatrix}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{bmatrix} a_{12} \begin{bmatrix}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{bmatrix} \\
a_{21} \begin{bmatrix}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{bmatrix}
a_{22} \begin{bmatrix}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{bmatrix}
\end{bmatrix} </math>