We have expressed the tensor commutation matrix n ⊗ n as linear combination of the tensor products of the generalized Gell-Mann matrices. The tensor commutation matrices 3 ⊗ 2 and 2 ⊗ 3 have been expressed in terms of the classical Gell-Mann matrices and the Pauli matrices.
In this paper, we prove that if the matrix of the linear system is symmetric, the Cholesky decomposition can be obtained from the Gauss elimination method without pivoting, without proving that the matrix of the system is positive definite.
A sign operator of energy, analogous to the helicity operator, but in the direction of what we call energy vector has been introduced. It is possible that there may be physical phenomena where energy vector should be considered. However, to write a wave function this energy vector needs a time vector. But, unlike the energy vector the time vector has no physical meaning yet. To make physical senses of the components of the time vector, the time dilation in special relativity has been studied and also the components of the time vector have been related to the tunneling times when an electron crosses a potential barrier. Physical results for quantum tunneling time will not be limited to this study.
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