Rules of Tensor and Matrix Operation for Liutex Calculation

Abstract

Vectors and tensors, which are widely used in physics, have their clear definitions and rules of calculations. However, in the practical applications and the process of derivations, vectors and tensors are usually expressed in the format of matrices which do not have as clear as their original mathematical definitions. The first problem is that there is no agreement on whether to use a column or row matrix to represent a vector which also leads to the problem of how to express a second-order tensor in a matrix format. In a word, the corresponding relations between vectors, tensors, and matrices are not very clear. Another problem is how to express vectors/tensors operation in matrices operations. It is clear that the first column matrix needs to be transposed before doing matrices multiplication when it is used to represent inner products of vectors. However, when it comes to the operation that a tensor dot product is a vector, some people do not transpose the first matrix (representing the second-order tensor). Liutex is a vortex identification concept that relates many vector and tensor operations. Therefore, it is important to have clear rules for using matrices to represent vectors and tensors. To overcome these problems and keep consistency, this paper provides rules of tensor and matrix operation for Liutex calculation. First, column matrices are used to represent vectors. Second, the matrix forms of vector/tensor operations are provided.