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Galbi
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Homework Statement
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.
jambaugh said: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,[itex] e^{11},e^{12},e^{21},e^{22}[/itex] where [itex] e^{ij}=e^i\otimes e^j[/itex]. Then the rank 4 tensor has matrix form:
[tex]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][/tex]
Invert that matrix and you have the "inverse" in the same basis.
jambaugh said: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,[itex] e^{11},e^{12},e^{21},e^{22}[/itex] where [itex] e^{ij}=e^i\otimes e^j[/itex]. Then the rank 4 tensor has matrix form:
[tex]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][/tex]
Invert that matrix and you have the "inverse" in the same basis.
jambaugh said: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,[itex] e^{11},e^{12},e^{21},e^{22}[/itex] where [itex] e^{ij}=e^i\otimes e^j[/itex]. Then the rank 4 tensor has matrix form:
[tex]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][/tex]
Invert that matrix and you have the "inverse" in the same basis.
A 4th order tensor is a mathematical object that has 4 indices and can be represented by a 4-dimensional array. It is used to describe the relationship between two sets of vectors or tensors in multidimensional space.
The inverse of a 4th order tensor is a tensor that, when multiplied with the original tensor, results in the identity tensor. It can be thought of as the "reciprocal" of the original tensor.
The inverse of a 4th order tensor can be calculated using various methods, depending on the specific properties of the tensor. One common method is the use of matrix inversion, where the tensor is converted into a matrix and then inverted using standard matrix inversion techniques.
The inverse of a 4th order tensor has various applications in physics, engineering, and mathematics. It is used in the study of fluid dynamics, electromagnetism, and elasticity, among others. It is also used in the development of computational models for complex systems.
Yes, there are limitations to using the inverse of a 4th order tensor. One limitation is that the inverse may not exist for certain tensors, such as those with zero determinants. Additionally, the calculation of the inverse can be computationally intensive, making it difficult to use in some applications.