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Generalized Gell-Mann matrices
[edit]Construction
[edit]Let be the matrix with 1 in the -th entry and 0 elsewhere. Consider the space of complex matrices, , for a fixed d. Define the following matrices
- For , .
- For , .
- Let , the identity matrix.
- For , .
- For , .
The collection of matrices defined above are called the generalized Gell-Mann matrices, in dimension d.
Properties
[edit]The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices. One can also check that they are orthogonal in the Hilbert-Schmidt inner product on . By the dimension count, we see that they span the vector space of complex matrices.
In dimensions 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.