Adjoint of a linear map

Andrew Thangaraj

Aug-Nov 2020

Recap

  • Vector space \(V\) over a scalar field \(F\)
    • \(F\): real field \(\mathbb{R}\) or complex field \(\mathbb{C}\) in this course
  • \(m\times n\) 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 \(v\): \(Tv=\lambda v\)
    • Some linear maps are diagonalizable
  • Inner products, norms, orthogonality and orthonormal basis
    • Upper triangular matrix for a linear map over an orthonormal basis
    • Orthogonal projection gives closest vector in the subspace
    • Least squares solution to a linear equation is orthogonal projection

Recall: Linear functionals and inner product

\(V\): finite-dimensional inner product space over \(F=\mathbb{R}\) or \(\mathbb{C}\)

\(\phi:V\to F\) is a linear functional

Riesz representation theorem: There exists unique \(u\in V\) such that \[\phi(v)=\langle v,u\rangle\]

How to find \(u\) for a given \(\phi\)?

Orthonormal basis for \(V\): \(e_1,\ldots,e_n\)

\(u=\overline{\phi(e_1)}e_1+\cdots+\overline{\phi(e_n)}e_n\)

Linear functionals from a linear map

\(V,W\): finite-dimensional inner product space over \(F=\mathbb{R}\) or \(\mathbb{C}\)

\(T:V\to W\) is a linear map

Fix some \(w\in W\)

Let \(\phi_{T,w}(v)=\langle Tv,w\rangle\)

\(\phi_{T,w}:V\to F\) is a linear functional

Riesz: There exists unique \(u_{T,w}\in V\) s.t. \[\phi_{T,w}(v)=\langle Tv,w\rangle=\langle v,u_{T,w}\rangle\]

Example

\(x=(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\)

\(Tx=(x_1+2x_2+3x_3+4x_4,3x_1+4x_2+5x_3+6x_4)\in\mathbb{R}^2\)

  1. \(w=(1,2)\)

\(\langle Tx,w\rangle=7x_1+10x_2+13x_3+16x_4\)

\(u_{T,w}=(7,10,13,16)\)

  1. \(w=(w_1,w_2)\)

\(\langle Tx,w\rangle=(w_1+3w_2)x_1+(2w_1+4w_2)x_2+(3w_1+5w_2)x_3+(4w_1+6w_2)x_4\)

\(u_{T,w}=(w_1+3w_2,2w_1+4w_2,3w_1+5w_2,4w_1+6w_2)\in\mathbb{R}^4\)

Adjoint of a linear map

\(V,W\): finite-dimensional inner product space over \(F=\mathbb{R}\) or \(\mathbb{C}\)

\(T:V\to W\) is a linear map

For \(w\in W\), let \(u_{T,w}\in V\) be s.t.

\[\langle Tv,w\rangle=\langle v,u_{T,w}\rangle\]

The adjoint of \(T\), denoted \(T^*\), is the function from \(W\) to \(V\) mapping \(w\) to \(u_{T,w}\)

\(T^*\): adjoint of \(T\) satisfies

\[\langle Tv,w\rangle=\langle v,T^*w\rangle\]

Example

\(x=(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\)

\(Tx=(x_1+2x_2+3x_3+4x_4,3x_1+4x_2+5x_3+6x_4)\in\mathbb{R}^2\)

\(w=(w_1,w_2)\in\mathbb{R}^2\)

\(\langle Tx,w\rangle=(w_1+3w_2)x_1+(2w_1+4w_2)x_2+(3w_1+5w_2)x_3+(4w_1+6w_2)x_4\)

\(T^*w=(w_1+3w_2,2w_1+4w_2,3w_1+5w_2,4w_1+6w_2)\in\mathbb{R}^4\)

Adjoint is a linear map

\(V,W\): finite-dimensional inner product space over \(F=\mathbb{R}\) or \(\mathbb{C}\)

\(T:V\to W\) is a linear map

\(T^*\): adjoint of \(T\) is a linear map from \(W\) to \(V\)

Proof

\(\begin{align} \langle v,T^*(w_1+w_2)\rangle &= \langle Tv,w_1+w_2\rangle\\ &=\langle Tv,w_1\rangle+\langle Tv,w_2\rangle\\ &=\langle v,T^*w_1\rangle+\langle v,T^*w_2\rangle\\ &=\langle v,T^*w_1+T^*w_2\rangle \end{align}\)

\(\begin{align} \langle v,T^*(aw)\rangle &= \langle Tv,aw\rangle\\ &=\bar{a}\langle Tv,w\rangle\\ &=\bar{a}\langle v,T^*w\rangle\\ &=\langle v,aT^*w\rangle \end{align}\)