Classification of Operators

Aug-Nov 2020

Recap

• 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 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 $\langle Tu,Tv\rangle=\langle u,v\rangle$