Classification of Operators

Andrew Thangaraj

Aug-Nov 2020


  • Vector space \(V\) over a scalar field \(F= \mathbb{R}\) or \(\mathbb{C}\)
  • \(m\times n\) matrix A represents a linear map \(T:F^n\to F^m\)
    • dim null \(T+\) dim range \(T=\) dim \(V\)
    • Solution to \(Ax=b\) (if it exists): \(u+\) null\((A)\)
  • Four fundamental subspaces of a matrix
    • Column space, row space, null space, left null space
  • Eigenvalue \(\lambda\) and Eigenvector \(v\): \(Tv=\lambda v\)
    • There is a basis w.r.t. which a linear map is upper-triangular
    • If there is a basis of eigenvectors, linear map is diagonal w.r.t. it
  • Inner products, norms, orthogonality and orthonormal basis
    • There is an orthonormal basis w.r.t. which a linear map is upper-triangular
    • Orthogonal projection: distance from a subspace
  • Adjoint of a linear map: \(\langle Tv,w\rangle=\langle v,T^*w\rangle\)
    • null \(T=\) \((\)range \(T^*)^{\perp}\)
  • Self-adjoint: \(T=T^*\), Normal: \(TT^*=T^*T\)
  • Complex/real spectral theorem: \(T\) is normal/self-adjoint \(\leftrightarrow\) orthonormal basis of eigenvectors
  • Positive operators: self-adjoint with non-negative eigenvalues
  • Isometries: normal with absolute value 1 eigenvalues

Operators and their null/range spaces

\(V\): inner product space, dim \(V=n\), \(T:V\to V\)

Type Property
Any dim null \(T\) + dim range \(T = n\)
null \(T=\) \((\)range \(T^*)^{\perp}\)
dim range \(T=\) dim range \(T^*\)
Upper-triangular matrix w.r.t. orthonormal basis
Invertible dim null \(T=0\), dim range \(T=n\)
Diagonalizable No special property
Normal null \(T=\) null \(T^*\), range \(T=\) range \(T^*\)
null \(T=\) \((\)range \(T)^{\perp}\)

Operators and their eigenvalues/eigenvectors

Operators and norms/inner products

  • (In a complex space) Self-adjoint iff \(\langle Tv,v\rangle\) is real

  • Normal iff \(\lVert Tv\rVert=\lVert T^*v\rVert\)

  • Isometry iff \(\langle Tu,Tv\rangle=\langle u,v\rangle\)