Eigenvalues of Hermitian and Unitary Matrices


Hermitian Matrices

If H is a hermitian matrix (i.e. H* = H - symmetric if real) then all the eigenvalues of H are real.

#{Corollary}: &exist. unitary matrix V such that V^{&minus.1}HV is a real diagonal matrix.

Unitary Matrices

If U is a unitary matrix ( i.e. U*U = I - orthonormal if real) the the eigenvalues of U have unit modulus.

#{Corollary}: &exist. unitary matrix V such that V^{&minus.1}UV is a diagonal matrix, with the diagonal elements having unit modulus.

Note: The columns of V are eigenvectors of the original matrix, so for hermitian and unitary matrices the eigenvectors can be chosen so as to form and orthonormal set.
If H is a real hermitian, i.e. symmetric matrix, it is similar to a real diagonal matrix and its eigenvectors may be chosen so as to form the columns of a (real) orthonormal (i.e. unitary) matrix. _ so &exist. orthonormal O such that

O*HO _ = _ matrix{&lamda._1, ... ,0/:, ... ,:/0, ... ,&lamda._{~n}}

Skew-Hermitian Matrices

An ~n # ~n matrix K is _ #{~{skew-hermitian}} _ if _ K* = &minus.K .
[ i.e. ~k_{~i ~j} = &minus.${~k}_{~j ~i} , _ or _ ~k_{~i ~j} = ~a_{~i ~j} + #{~i}~b_{~i ~j} = &minus.~a_{~j ~i} + #{~i}~b_{~j ~i} ]




Skew-Symmetric Matrices

An ~n # ~n real matrix S is _ #{~{skew-symmetric}} _ if _ S^T = &minus.S .
[ i.e. ~s_{~i ~j} = &minus.~s_{~j ~i} _ &forall. ~i, ~j , _ &imply. _ ~s_{~i ~i} = 0 _ &forall. ~i ]




Eigenvalues of Skew-Hermitian Matrices

All the eigenvalues of an ~n # ~n skew-hermitian matrix K are pure imaginary.
Further if ~n is even then &vdash.K&vdash. is real, _ if ~n is odd then &vdash.K&vdash. is imaginary or zero.

Unitary Decomposition

If U is an ~n # ~n unitary matrix with no eigenvalue = &pm.1, _ then &exist. an ~n # ~n skew-hermitian matrix K such that

U _ = _ &pm.1(K + I)(K &minus. I)^{&minus.1} ,

and conversely.