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Generalized Gell-Mann matrices

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Construction

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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

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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.