Spectral Theorem

• The Spectral Theorem states that for any normal (in ) or self-adjoint (in ) linear operator, there exists a simpler canonical form. The canonical form is obtained by decomposing the linear operator as a series of projections onto orthogonal eigenspaces.

• (Friedberg 6.24) Spectral Theorem. Suppose that is a linear operator on a finite dimensional inner product space over . Assume that is normal if and is self-adjoint if .

  If are the distinct eigenvalues of , let be the eigenspace of corresponding to eigenvalue and let be the orthogonal projection on . Then

  • If denotes the direct sum of the subspaces of , , then
  • for
  • . This decomposition is called the resolution of the identity operator induced by
  • . The decomposition of this sum is called the spectral decomposition.
  • The spectral decomposition is unique.
  • If is the union of orthonormal bases of the s and then

• (Friedberg 6.24.1) If then is normal if and only if for some polynomial .

• (Friedberg 6.24.2) If , then is unitary if and only if is normal and for every

• (Friedberg 6.24.3) If and is normal, then is self-adjoint if and only if every eigenvalue of is real.

• (Friedberg 6.24.4) Let have a spectral decomposition of . Then each is a polynomial in

• (Friedberg e6.6.7) Let be a normal operator and a linear operator on a finite dimensional complex inner product space . Let be a spectral decomposition of .

  • If is a polynomial then
  • If for some then
  • commutes with if and only if commutes with each
  • If is normal and commutes with then if we have as (not necessarily distinct) eigenvalues of ,
  • There exists a normal operator on such that
  • is invertible if
  • is a projection if and only if every eigenvalue of is or .
  • if and only if every is an imaginary number.

Links §

• [[Linear Algebra by Friedberg Insel and Spence|Friedberg, Insel and Spence]/]