Double Dot Product

The double dot product is an important concept of mathematical algebra. It is a way of multiplying the vector values. Before learning a double dot product we must understand what is a dot product. 

Stating it in one paragraph, Dot products are one method of simply multiplying or even more vector quantities. Two vector’s dot product produces a scalar number. As a result, the dot product of two vectors is often referred to as a scalar. It is also the vector sum of the adjacent elements of two numeric values in sequence. Dimensionally, it is the sum of two vectors’ Euclidean magnitudes as well as the cos of such angles separating them. The dot product’s vector has several uses in mathematics, physics, mechanics, and astrophysics. 

In this article, we will also come across a word named tensor. Let us describe what is a tensor first. A tensor is a three-dimensional data model. Matrices and vectors constitute two-dimensional computational models and one-dimensional computational models or data structures, respectively. Tensors are identical to some of these record structures on the surface, but the distinction is that they could occur on a dimensionality scale from 0 to n.

We must also understand the rank of the tensors we’ll come across.

The set of orientations (and therefore the dimensions of the collection) is designed to understand a tensor to determine its rank (or grade). For instance, characteristics requiring just one channel (first rank) may be fully represented by a 31 dimensional array, but qualities requiring two directions (second class or rank tensors) can be entirely expressed by 9 integers, as a 33 array or the matrix. As a result, an nth ranking tensor may be characterised by 3n components in particular.

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What is a double dot product?

The double dot combination of two values of tensors is the shrinkage of such algebraic topology with regard to the very first tensor’s final two values and the subsequent tensor’s first two values. It is a matter of tradition such contractions are performed or not on the closest values. In this article, I’ll discuss how this decision has significant ramifications.

Let us understand it in detail.

Consider, m and n to be two second rank tensors, To define these into the form of a double dot product of two tensors m:n we can use the following methods.

1st method: m:n = mijnji

2nd method: m:n = mijnij

Several 2nd ranked tensors (stress, strain) in the mechanics of continuum are homogeneous, therefore both formulations are correct. Nevertheless, in the broader situation of uneven tensors, it is crucial to examine which standard the author uses. In this post, we will look at both concepts in turn and see how they alter the formulation of the transposition of 4th ranked tensors, which would be the first description remembered.

What is a 4th rank tensor transposition or transpose?

Consider A to be a fourth-rank tensor. Its continuous mapping tens xA:x(where x is a 3rd rank tensor) is hence an endomorphism well over the field of 2nd rank tensors. As a result, its inversion or transposed ATmay be defined, given that the domain of 2nd ranked tensors is endowed with a scalar product (.,.). So, by definition,

AT:x,y=x,A:y

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Defining the scalar product of 2nd ranked tensors as

x,y=xijyij

as a result of which the scalar product of 2 2nd ranked tensors is strongly connected to any notion with their double dot product Any description of the double dot product yields a distinct definition of the inversion, as demonstrated in the following paragraphs.

First Definition

Let us have a look at the first mathematical definition of the double dot product.

Consider two double ranked tensors or the second ranked tensors given by,

x:y=xijyji

x,y=xT:y

Also, consider A as a fourth ranked tensor quantity. Finding the components of AT

x,A:y=xT:(A:y)=(xT)ji(A:y)ij=xijAijklylk

Defining the A’ which is a fourth ranked tensor component-wise as A’ijkl=Alkji

x,A:y=ylkA’lkjixij=(yt)kl(A’:x)lk=yT:(A’:x)=A’:x,y

The equation we just made defines or proves that A’s transposition is A’. Now, if we use the first definition then any 4th ranked tensor quantity’s components will be as,

(AT)ijkl=Alkji

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Second Definition

Considering the second definition of the double dot product,

x:y=xijyij

Similar to the first definition x and y is 2nd ranked tensor quantities,

Therefore,

x,y=xijyij

 also, consider A as a 4th ranked tensor. Again if we find AT’s component, it will be as,

x,A:y=x:(A:y)=(x)ij(A:y)ij=xijAijklykl

Again bringing a fourth ranked tensor defined by A’’. Whose component-wise definition is as,

A”ijkl=Aklij

Equations becomes as,

x,A:y=yklA”klijxij=(y)kl(A”:x)kl=y:(A”:x)=A”:x,y

The equation we just fount detemrines that A’s transposition os A’’’. Using the second definition a 4th ranked tensors component’s transpose will be as,

(AT)ijkl=Aklij

Conclusion

To sum up, A dot product is a simple multiplication of two vector values and a tensor is a 3d data model structure. The rank of a tensor scale from 0 to n depends on the dimension of the value. Two tensor’s double dot product is a contraction of the last two digits of the two last digits of the first tensor value and the two first digits of the second or the coming tensor value. 

There are two definitions for the transposition of the double dot product of the tensor values that are described above in the article.