Skew-Hermitian

The matrix W1 K is in general skew-Hermitian due to Eq. (10), and hence its diagonal elements w P(R J are pure imaginary quantities. If we require that the /f,ad be real, then the matrix W ad becomes real and skew-symmetric with the diagonal elements equal to zero and the off-diagonal elements satisfying the relation... 

These coupling matrix elements are scalars due to the presence of the scalar Laplacian in Eq. (25). These elements are, in general, complex but if we require the /)L id to be real they become real. The matrix Wl-2 ad(R-/.), unlike its first-derivative counterpart, is neither skew-Hermitian nor skew-symmetric. 

Any linear transformation A satisfying the condition A + A =0 can be called anti-Hermitian or skew-Hermitian. 

Adjoint Orthogonal Unitary Hermitian Skew-Hermitian Normal Permutation, Pseudo-permutation Af At = (AT) AtA = E AfA = E Af = A Af = -A AA+ = AfA see text... 

Therefore, the eigenvalues of a Hermitian matrix are real, and the eigenvalues of a skew-Hermitian matrix are pure imaginary. Now consider the eigenvectors x) and x ) belonging to two different eigenvalues a, a of a self-adjoint matrix A. 

The coefficient at A describes the linear response of the quantity A to the perturbation W. It can be given a rather more symmetric form. Indeed the amplitude of the j-th unperturbed state in the correction to the fc-th state is proportional to some skew Hermitian operator (the perturbation matrix W is Hermitian, but the denominator changes its sign when the order of the subscripts changes). With this notion and assuming that Wkk = 0 (see above) we can remove the restriction in the summation and write ... 

Figure 3 shows a fit of (2.19) to computational results generated using a non-Hermitian T operator, such that its Hermitian and skew-Hermitian parts have the same variance. 

The compact form Gc is constructed here as follows. Consider an involution T G G,tA = —ATf where T implies transposition. The fixed points of this involution are skew-Hermitian matrices with a zero trace. It is readily seen that this space is the Lie algebra of the compact group su(n). Indeed, calculating the Killing form on this real form Go, we immediately obtain that it is negative definite. 

Gn = (the normal noncompact form) = su(n) O sl(n, R) = so(n), because skew-Hermitian real matrices are skew-symmetric. 

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