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The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σ i, so that the Hermitian matrix is written as a Pauli vector. [2] In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as ...
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli. The Pauli group on n {\displaystyle n} qubits, G n {\displaystyle G_{n}} , is the group generated by the operators described above applied to each of n {\displaystyle n} qubits in the tensor product Hilbert space ( C 2 ) ⊗ n {\displaystyle ...
In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the
More compactly, = , and = , where denotes the Kronecker product and the (for j = 1, 2, 3) denote the Pauli matrices. In addition, for discussions of group theory the identity matrix (I) is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrices
The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3). The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space.
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.