A four-vector A can also be defined in using the Pauli matrices as a basis, again in various equivalent notations: A = ( A 0 , A 1 , A 2 , A 3 ) = A 0 σ σ 0 + A 1 σ σ 1 + A 2 σ σ 2 + A 3 σ σ 3 = A 0 σ σ 0 + A i σ σ i = A α α σ σ α α or explicitly: A = A 0 ( 1 0 0 1 ) + A 1 ( 0 1 1 0 ) + A 2 ( 0 − − i i 0 ) + A 3 ( 1 0 0 − − 1 ) = ( A 0 + A 3 A 1 − − i A 2 A 1 + i A 2 A 0 − − A 3 ) and in this formulation, the four-vector is represented as a Hermitian matrix (the matrix transpose and complex conjugate of the matrix leaves it unchanged), rather than a real-valued column or row vector. The determinant of the matrix is the modulus of the four-vector, so the determinant is an invariant: | A | = | A 0 + A 3 A 1 − − i A 2 A 1 + i A 2 A 0 − − A 3 | = ( A 0 + A 3 ) ( A 0 − − A 3 ) − − ( A 1 − − i A 2 ) ( A 1 + i A 2 ) = ( A 0 ) 2 − − ( A 1 ) 2 − − ( A 2 ) 2 − − ( A 3 ) 2