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:

$\bigotimes _{i=1}^{n}T_{i}=T_{1}\otimes T_{2}\otimes \dots \otimes T_{n}$

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:

$T_{1}\otimes T_{2}\neq T_{2}\otimes T_{1}$

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

$\bigotimes _{i=1}^{n}(T_{i}+T'_{i})=\bigotimes _{i=1}^{n}T_{i}+\bigotimes _{i=1}^{n}T'_{i}$

• 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 $d_{i}$:

${\text{dim}}(\bigotimes _{i=1}^{n}T_{i})=\prod _{i=1}^{n}{\text{dim}}(T_{i})$

Examples

Example 1: Simple Tensotory of Vectors

Consider two vectors ${\vec {v}}=[v_{1},v_{2}]$ and ${\vec {w}}=[w_{1},w_{2}]$. Their tensor product is:

${\vec {v}}\otimes {\vec {w}}={\begin{bmatrix}v_{1}\cdot w_{1}&v_{1}\cdot w_{2}\\v_{2}\cdot w_{1}&v_{2}\cdot w_{2}\end{bmatrix}}$

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

$\bigotimes _{i=1}^{3}{\vec {v}}_{i}={\vec {v}}\otimes {\vec {w}}\otimes {\vec {u}}$

resulting in a three-dimensional tensor.

Example 2: Tensotory of Matrices

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

$A\otimes B={\begin{bmatrix}a\cdot B&b\cdot B\\c\cdot B&d\cdot B\end{bmatrix}}={\begin{bmatrix}a\cdot e&a\cdot f&b\cdot e&b\cdot f\\a\cdot g&a\cdot h&b\cdot g&b\cdot h\\c\cdot e&c\cdot f&d\cdot e&d\cdot f\\c\cdot g&c\cdot h&d\cdot g&d\cdot h\end{bmatrix}}$

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.