Draft:Tensotory

Tensotory

In mathematics and theoretical physics, a tensotory refers to the iterated application of the Tensor product over a finite or infinite sequence of tensors. It is an operation that generalizes the idea of aggregation of tensors in higher-dimensional spaces. The term "tensotory" is a neologism coined to describe this process in a manner analogous to "summatory" (for sums) and "productory" (for products).

Definition

Given a sequence of tensors $T_1,T_2,\dots ,T_n$, the tensotory is defined as the iterated tensor product:

$\underset{i=1}{\overset{n}{⨂}}{T}_i=T_1\otimes T_2\otimes \dots \otimes T_n$

Here, $\otimes$ denotes the tensor product operator, and the result is a higher-dimensional tensor whose rank is the sum of the ranks of the individual tensors.

Properties

• Associativity: The tensor product is associative:

$(T_1\otimes T_2)\otimes T_3=T_1\otimes (T_2\otimes T_3)$ Thus, the order of operations in a tensotory does not affect the result.

• Non-commutativity: The tensor product is generally not commutative:

$T_1\otimes T_2\ne T_2\otimes T_1$

• Linearity: The tensotory is linear with respect to the addition of tensors:

$\underset{i=1}{\overset{n}{⨂}}(T_i+T{\text{'}}_i)=\underset{i=1}{\overset{n}{⨂}}{T}_i+\underset{i=1}{\overset{n}{⨂}}T{\text{'}}_i$

• Dimensionality: If $T_i$ are tensors of dimensions $d_i$, the resulting tensor from the tensotory will have a dimension equal to the product of $d_i$:

$\text{dim}(\underset{i=1}{\overset{n}{⨂}}{T}_i)=\prod _{i=1}^n\text{dim}(T_i)$

Examples

Example 1: Simple Tensotory of Vectors

Consider two vectors $\overrightarrow{v}=[v_1,v_2]$ and $\overrightarrow{w}=[w_1,w_2]$. Their tensor product is:

$\overrightarrow{v}\otimes \overrightarrow{w}=\left[\begin{array}{cc}{v}_1\cdot w_1& v_1\cdot w_2\\ v_2\cdot w_1& v_2\cdot w_2\end{array}\right]$

Extending this to a tensotory of three vectors $\overrightarrow{v},\overrightarrow{w},\overrightarrow{u}$, we compute:

$\underset{i=1}{\overset{3}{⨂}}{\overrightarrow{v}}_i=\overrightarrow{v}\otimes \overrightarrow{w}\otimes \overrightarrow{u}$

resulting in a three-dimensional tensor.

Example 2: Tensotory of Matrices

Given two matrices $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]$, their tensor product is:

$A\otimes B=\left[\begin{array}{cc}a\cdot B& b\cdot B\\ c\cdot B& d\cdot B\end{array}\right]=\left[\begin{array}{cccc}a\cdot e& a\cdot f& b\cdot e& b\cdot f\\ a\cdot g& a\cdot h& b\cdot g& b\cdot h\\ c\cdot e& c\cdot f& d\cdot e& d\cdot f\\ c\cdot g& c\cdot h& d\cdot g& d\cdot h\end{array}\right]$

A tensotory over multiple matrices $A_1,A_2,\dots ,A_n$ follows this same pattern, producing a higher-dimensional structure.

Applications

• Quantum Computing: Tensor products are foundational for representing quantum states in composite systems.
• Machine Learning: Tensories are used to represent high-dimensional data and features in deep learning architectures.
• Physics: Tensories describe complex systems, including stress-strain relationships in materials and general relativity.

See Also

• Tensor
• Tensor product
• Summation
• Product notation

External Links

• MathWorks - Tensor Product Documentation
• Wikipedia article on Tensor
• Video explaining tensor operations

References

• "Mathematical Methods in the Physical Sciences" by Mary L. Boas. John Wiley & Sons, 2006.
• Kreyszig, E. (2011). "Advanced Engineering Mathematics" (10th ed.). Wiley. ISBN 978-0470458365.