Chapter 7: Self-Adjoint and Normal Operators

What is a self-adjoint operator?

A linear map T:V\rightarrow V is self-adjoint (or Hermitian) if TT=T^* (i.e.\langle T(v),w\rangle=\langle vT(w)\rangle\forall v,w\in V). In matrix terms, this means A=A^{\dagger}(conjugate transpose).

What is a normal operator?

T:V\rightarrow V is normal if it commutes with its adjoint TT^*=T^*T. Examples: self-adjoint, unitary, and skew-adjoint operators are all normal.

Spectral Theorem for normal operators (in complex space)

If T is normal on a finite-dimensional complex inner product space, then:

• There exists an orthonormal basis of eigenvectors.

• T is unitarily diagonalizable.

Spectral Theorem for self-adjoint operators (in real space)

If T is self-adjoint on a real inner product space, then:

• All eigenvalues are real

• There exists an orthonormal basis of eigenvectors

• T is orthogonally diagonalizable

What is the geometric meaning of self-adjoint operators?

Self-adjoint transformations stretch/compress along orthogonal directions, and never “rotate” vectors. They preserve symmetry and have real eigenvalues.

What is a skew-adjoint operator?

A linear operator T such that T^*=-T. These generate rotations in real space (e.g. 2\times 2 rotation matrices come from skew-symmetric generators).

Show that T=\left[\begin{array}{cc}0&-1\\1&0\end{array}\right] is skew-adjoint.

Its transpose is T^T=\left[\begin{array}{cc}0&1\\-1&0\end{array}\right]=-T. Therefore, it is skew-adjoint. Geometrically, this is a rotation by 90^\circ.