11.5: Positive operators
- Page ID
- 309
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Recall that self-adjoint operators are the operator analog for real numbers. Let us now define the operator analog for positive (or, more precisely, nonnegative) real numbers.
Definition 11.5.1. An operator \(T\in \mathcal{L}(V)\) is called positive (denoted \(T\ge 0\)) if \(T=T^*\) and \(\inner{Tv}{v} \ge 0\) for all \(v\in V\).
If \(V\) is a complex vector space, then the condition of self-adjointness follows from the condition \(\inner{Tv}{v} \ge 0\) and hence can be dropped.
Example 11.5.2. Note that, for all \(T \in \mathcal{L}(V)\), we have \(T^*T\ge 0\) since \(T^*T\) is self-adjoint and \(\inner{T^*Tv}{v}=\inner{Tv}{Tv} \ge 0\).
Example 11.5.3. Let \(U\subset V\) be a subspace of \(V\) and \(P_U\) be the orthogonal projection onto \(U\).
Then \(P_U\ge 0\). To see this, write \(V=U \oplus U^\bot\) and \(v=u_v+u_v^\bot\) for each \(v\in V\), where \(u_v \in U\) and \(u_v^\bot \in U^\bot\). Then \(\inner{P_U v}{w} = \inner{u_v}{u_w+u_w^\bot} = \inner{u_v}{u_w} = \inner{u_v+u_v^\bot}{u_w} = \inner{v}{P_U w}\) so that \(P_U^*=P_U\). Also, setting \(v=w\) in the above string of equations, we obtain \(\inner{P_U v}{v}=\inner{u_v}{u_v} \ge 0\), for all \(v\in V\). Hence, \(P_U\ge 0\).
If \(\lambda\) is an eigenvalue of a positive operator \(T\) and \(v\in V\) is an associated eigenvector, then \(\inner{Tv}{v} = \inner{\lambda v}{v} = \lambda \inner{v}{v} \ge 0\). Since \(\inner{v}{v}\ge 0\) for all vectors \(v\in V\), it follows that \(\lambda\ge 0\). This fact can be used to define \(\sqrt{T}\) by setting
\begin{equation*}
\sqrt{T} e_i = \sqrt{\lambda_i} e_i,
\end{equation*}
where \(\lambda_i\) are the eigenvalues of \(T\) with respect to the orthonormal basis \(e=(e_1,\ldots,e_n)\). We know that these exist by the Spectral Theorem.
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