# Inverse of 4th order tensor

1. Apr 1, 2015

### Galbi

1. The problem statement, all variables and given/known data
I'm looking for how to calculate inverse of the 4th order tensor. That is,
A:A-1=A-1:A=I(4)
If I know a fourth order tensor A, then how can I calculate A-1 ?
Let's just say it is inversible.

2. Relevant equations

3. The attempt at a solution

2. Apr 1, 2015

### jambaugh

By "fourth order" I presume you mean rank 4, but that is with regard to the original vector space. A tensor is also a "Vector" in its own vector space as well as a linear operator in several settings but you seem to be implying a double contraction wherein you are mapping rank 2 tensors to rank 2 tensors.

Your task then is to write down a basis for the rank 2 tensor space upon which the rank 4 tensor acts and express that rank 4 tensor's components in that basis in the form of a matrix. You then use the standard matrix inversion techniques to find its inverse.

Example for 2 dimensional vectors you would choose as a basis for your rank 2 space say,$e^{11},e^{12},e^{21},e^{22}$ where $e^{ij}=e^i\otimes e^j$. Then the rank 4 tensor has matrix form:
$$A=\left[\begin{array}{cccc} A_{1111} & A_{1112} & A_{1121} & A_{1122} \\ A_{1211} & A_{1212} & A_{1221} & A_{1222} \\ A_{2111} & A_{2112} & A_{2121} & A_{2122} \\ A_{2211} & A_{2212} & A_{2221} & A_{2222} \end{array}\right]$$

Invert that matrix and you have the "inverse" in the same basis.

3. Apr 1, 2015

### Galbi

Thank you very much.
Since I'm a novice at tensor calculation, It's hard to understand what you're saying.

If I have 3x3x3x3 tensors, then do the components of inversion of the matrix as seen below correspond to the components of inverse of A ?

$$A=\left[\begin{array}{cccc} A_{1111} & A_{1112} & A_{1113} & A_{1121}& A_{1122}& A_{1123}& A_{1131}& A_{1132}& A_{1133} \\ A_{1211} & A_{1212} & A_{1213} & A_{1221}& A_{1222}& A_{1223}& A_{1231}& A_{1232}& A_{1233} \\ A_{1311} & A_{1312} & A_{1313} & A_{1321}& A_{1322}& A_{1323}& A_{1331}& A_{1332}& A_{1333} \\ A_{2111} & A_{2112} & A_{2113} & A_{2121}& A_{2122}& A_{2123}& A_{2131}& A_{2132}& A_{2133} \\ A_{2211} & A_{2212} & A_{2213} & A_{2221}& A_{2222}& A_{2223}& A_{2231}& A_{2232}& A_{2233} \\ A_{2311} & A_{2312} & A_{2313} & A_{2321}& A_{2322}& A_{2323}& A_{2331}& A_{2332}& A_{2333} \\ A_{3111} & A_{3112} & A_{3113} & A_{3121}& A_{3122}& A_{3123}& A_{3131}& A_{3132}& A_{3133} \\ A_{3211} & A_{3212} & A_{3213} & A_{3221}& A_{3222}& A_{3223}& A_{3231}& A_{3232}& A_{3233} \\ A_{3311} & A_{3312} & A_{3213} & A_{3321}& A_{3322}& A_{3323}& A_{3331}& A_{3332}& A_{3333} \\ \end{array}\right]$$

I'm going to check this with MATLAB now.
Thank you again!

4. Apr 1, 2015

### Galbi

Dear jambaugh,

It works! Thank you very much!!!!