8.6E: The Singular Value Decomposition Exercises
- Page ID
- 132842
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercises for 1
solutions
2
If \(ACA=A\) show that \(B=CAC\) is a middle inverse for \(A\).
For any matrix \(A\) show that
\[\Sigma_{A^{T}}=(\Sigma_{A})^{T} \nonumber \]
If \(A\) is \(m\times n\) with all singular values positive, what is \(rank \;A\)?
If \(A\) has singular values \(\sigma_{1},\dots ,\sigma_{r}\), what are the singular values of:
\(A^{T}\) \(tA\) where \(t>0\) is real \(A^{-1}\) assuming \(A\) is invertible.
- \(t\sigma _{1},\dots ,t\sigma _{r}.\)
If \(A\) is square show that \(| \det A |\) is the product of the singular values of \(A\).
If \(A\) is square and real, show that \(A=0\) if and only if every eigenvalue of \(A^T A\) is \(0\).
Given a SVD for an invertible matrix \(A\), find one for \(A^{-1}\). How are \(\Sigma_{A}\) and \(\Sigma_{A^{-1}}\) related?
If \(A=U\Sigma V^{T}\) then \(\Sigma\) is invertible, so \(A^{-1}=V\Sigma^{-1}U^{T}\) is a SVD.
Let \(A^{-1}=A=A^{T}\) where \(A\) is \(n\times n\). Given any orthogonal \(n\times n\) matrix \(U\), find an orthogonal matrix \(V\) such that \(A=U\Sigma_{A}V^{T}\) is an SVD for \(A\).
If \(A= \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]\) do this for:
\(U=\frac{1}{5} \left[ \begin{array}{rr} 3 & -4 \\ 4 & 3 \end{array} \right]\) \(U=\frac{1}{\sqrt{2}} \left[ \begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array} \right]\)
- First \(A^{T}A=I_{n}\) so \(\Sigma _{A}=I_{n}\).
\[\begin{aligned} A &=& \frac{1}{\sqrt{2}}\left[ \begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right] \frac{1}{\sqrt{2}} \left[ \begin{array}{rr} 1 & 1 \\ -1 & 1 \end{array} \right] \\ & =&\frac{1}{\sqrt{2}}\left[ \begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array} \right] \frac{1}{\sqrt{2}} \left[ \begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array} \right] \\ & =& \left[ \begin{array}{rr} -1 & 0 \\ 0 & 1 \end{array} \right] \end{aligned} \nonumber \]
Find a SVD for the following matrices:
\(A= \left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \\ 1 & 0 \end{array} \right]\) \(\left[ \begin{array}{rrr} 1 & 1 & 1 \\ -1 & 0 & -2 \\ 1 & 2 & 0 \end{array} \right]\)
- \[\hspace*{-5em} = \scriptsize \frac{1}{5}\left[ \begin{array}{rr} 3 & 4 \\ 4 & -3 \end{array} \right] \left[ \begin{array}{rrrr} 20 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \end{array} \right] \frac{1}{2} \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \end{array} \right] \nonumber \]
Find an SVD for \(A= \left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array} \right]\).
If \(A=U\Sigma V^{T}\) is an SVD for \(A\), find an SVD for \(A^{T}\).
Let \(A\) be a real, \(m\times n\) matrix with positive singular values \(\sigma_{1},\sigma_{2},\dots ,\sigma_{r}\), and write
\[s(x)=(x-\sigma_{1})(x-\sigma_{2})\cdots (x-\sigma_{r}) \nonumber \]
- Show that \(c_{A^{T}A}(x)=s(x)x^{n-r}\) and \(c_{A^{T}A}(c)=s(x)x^{m-r}\).
- If \(m\leq n\) conclude that \(c_{A^{T}A}(x)=s(x)x^{n-m}\).
If \(G\) is positive show that:
- \(rG\) is positive if \(r\geq 0\)
- \(G+H\) is positive for any positive \(H\).
- If \(\mathbf{x}\in \mathbb{R}^{n}\) then \(\mathbf{x}^{T}(G+H)\mathbf{x}=\mathbf{x}^{T}G\mathbf{x}+\mathbf{x}^{T}H\mathbf{x}\geq 0+0=0\).
If \(G\) is positive and \(\lambda\) is an eigenvalue, show that \(\lambda \geq 0\).
If \(G\) is positive show that \(G=H^{2}\) for some positive matrix \(H\). [Hint: Preceding exercise and Lemma [lem:svdlemma5]]
If \(A\) is \(n\times n\) show that \(AA^{T}\) and \(A^{T}A\) are similar. [Hint: Start with an SVD for \(A\).]
Find \(A^{+}\) if:
- \(A= \left[ \begin{array}{rr} 1 & 2 \\ -1 & -2 \end{array} \right]\)
- \(A= \left[ \begin{array}{rr} 1 & -1 \\ 0 & 0 \\ 1 & -1 \end{array} \right]\)
- \(\left[ \begin{array}{rrr} \frac{1}{4} & 0 & \frac{1}{4} \\ -\frac{1}{4} & 0 & -\frac{1}{4} \end{array} \right]\)
Show that \((A^{+})^{T}=(A^{T})^{+}\).