Manuals/calci/MATRIXTENSORPRODUCT

MATRIXTENSORPRODUCT (a,b)

• $a$ and $b$ are any two matrices.

Description

• This function shows the Tensor product of the matrix.
• In $MATRIXTENSORPRODUCT(a,b)$ , $a$ and $b$ are any two matrices.
• Here matrices $a$ and $b$ should be square matrix with same order.
• Tensor product is denoted by $\otimes$.
• 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.
• It is denoted by $V\otimes W$.
• The tensor product of V and W is the vector space generated by the symbols $v\otimes w$, with $v\in V$ and $w\in W$.
• The tensor product from the direct sum vector space, whose dimension is the sum of the dimensions of the two summands:

$dim(V\otimes W)=dimV+dimW$

• Now consider any 2x2 matrices:

${\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}\otimes {\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}}={\begin{bmatrix}a_{11}{\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}}a_{12}{\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}}\\a_{21}{\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}}a_{22}{\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}}\end{bmatrix}}$ = ${\begin{bmatrix}a_{11}b_{11}&a_{11}b_{12}&a_{12}b_{11}&a_{12}b_{12}\\a_{11}b_{21}&a_{11}b_{22}&a_{12}b_{21}&a_{12}b_{22}\\a_{21}b_{11}&a_{21}b_{12}&a_{22}b_{11}&a_{22}b_{12}\\a_{21}b_{21}&a_{21}b_{22}&a_{22}b_{21}&a_{22}b_{22}\end{bmatrix}}$