Inverting a Tensor - How to Find Out?

[tulip]

Thank you for reading.

If I have an object (is it correct to call it a tensor?) whose components are defined by:

$$X_{mikl}=(R^{-1})_{mi}R_{kl}-(S^{-1})_{mi}S_{kl},$$

where R and S are invertible matrices. I want to find the "inverse" of X, i.e. to find (X^{-1}) such that,

$$(X^{-1})_{qkpm}X_{mikl}=\delta_{lq}\delta_{pi}$$.

Is there a way to find out whether (X^{-1}) exists? A matrix doesn't have an inverse if its determinant is zero - is there a similar rule here? I've tried some trial functions for X^{-1} but nothing works, and I want to know whether there's a better way of tackling this problem.

 

[Fernsanz]

With no further constraints the tensor you have proposed is, in general, not invertible.

Consider for example the case $$R=S$$. Then $$X$$ will be the null tensor. Or even the less trivial case $$R=S^{-1}$$ will lead also to $$X=0$$ if $$R^2=1$$ (idempotent).

Therefore, in general, it is not invertible.

 

[tulip]

R is not equal to S, or to the inverse of S. It is clearly true that for special cases where X=0 there is no inverse, but I don't see how this tells us that X is generally non-invertible. Can you explain?

 

[Fernsanz]

If for the problem stated there are cases where X is not invertible (even if there is just one of such cases) then it can not exist a theorem, a result, or an algorithm which allows us to invert the given expression with generelity, i.e. regardless of the values of R and S, (i.e., operating with "letters" in full generality insted of with "numbers"). Hence, the invertibility or not of X will depend on the particular values of R and S, and so we say that "in general" X is not invertible.

So, the meaning of "in general X is not invertible" is not "X is non-invertible more often than not"; the meaning is actually "the invertibility of X has to be determined on a particular case basis, it can not be determined generally (no matter the values of R and S)"

 

[blue_raver22]

I can provide some insights on how to approach this problem of finding the inverse of a tensor. Firstly, it is important to note that tensors are mathematical objects that generalize the concepts of scalars, vectors, and matrices. They are used extensively in physics, engineering, and other fields to describe physical quantities that have magnitude and direction.

In order to find the inverse of a tensor X, we can use the same approach as finding the inverse of a matrix. As mentioned, the inverse of a matrix does not exist if its determinant is zero. Similarly, for tensors, the determinant-like quantity is called the trace. If the trace of X is zero, then the inverse of X does not exist.

However, even if the trace of X is non-zero, it does not necessarily mean that the inverse of X exists. In such cases, we can use a method called singular value decomposition (SVD) to determine if the inverse exists. SVD is a technique used to decompose a matrix or tensor into its constituent parts, making it easier to analyze and manipulate.

Another approach to finding the inverse of a tensor is to use the Moore-Penrose pseudoinverse. This is a generalization of the inverse of a matrix and can be used for tensors as well. The pseudoinverse of a tensor X is denoted as X^+ and is defined as the unique tensor that satisfies the following conditions:

1) X^+X=X
2) XX^+=I (identity tensor)

Using these conditions, we can find the pseudoinverse of X by solving a set of linear equations.

In conclusion, finding the inverse of a tensor can be a challenging task, but there are methods like SVD and pseudoinverse that can help us determine if the inverse exists and how to find it. It is important to understand the properties of tensors and use appropriate mathematical techniques to solve such problems.