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The kernel of this map, a matrix whose trace is zero, is often said to be traceless or trace free, and these matrices form the simple Lie algebra, which is the Lie algebra of the special linear group of matrices with determinant 1.
These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. [1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark ...
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.
In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1.. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case.
If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. The sum of two skew-symmetric matrices is skew-symmetric. A scalar multiple of a skew-symmetric matrix is skew-symmetric. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero.
The Lie algebra can be explicitly realized as a matrix Lie algebra of traceless matrices. This is the fundamental representation for s l n C {\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} } . Set M i , j {\displaystyle M_{i,j}} to be the matrix with one in the i , j {\displaystyle i,j} entry and zeroes everywhere else.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of (n 2 − 1)-dimensional Euclidean space. [1]