matrix norm equivalence
}\)Equality is attained for \(A = \left(
\newcommand{\FlaOneByTwo}[2]{
} \\ Then there exist positive scalars \(\sigma \) and \(\tau \) such that for all \(A \in \C^{m \times n} \)The proof again builds on the fact that the supremum over a compact set is achieved and can be replaced by the maximum.We will prove that there exists a \(\tau \) such that for all \(A \in \C^{m \times n} \)Let \(A \in \C^{m \times n} \) be an arbitary matrix. \end{array} \right) \text{. \newcommand{\DeltaA}{\delta\!\!A}
} \end{array} \right) \text{. ~~~\leq~~~~ \lt \sqrt{m} \| z \|_2 \geq \| z \|_1 \end{array} \| A \|_1 \\ \end{array} \\ \newcommand{\Rn}{\mathbb R^n}
\newcommand{\FlaOneByTwoSingleLine}[2]{
#1 \amp #2 \amp #3 \\ \hline = \max_{\vert x \vert_2 = 1} \| x \|_2 = 1.
#2 \\ \amp 1 \amp 1 \amp 1 \\ \begin{equation*}
} \\ \newcommand{\Ck}{\mathbb C^k} \newcommand{\deltay}{\delta\!y} \newcommand{\complexone}{ \frac{\vert \vert \vert A \vert \vert \vert}{\| A \|} \| A ~~~=~~~~ \lt \mbox{ definition } \gt }\)(See Homework 1.3.7.2, which requires the SVD, as mentioned...)\(
~~~ \begin{array}{l} }\)\(\| A \|_2 \leq \sqrt{m} \| A \|_\infty \text{:}\)Equality is attained for \(A = \left( \| A \|_F \leq \sqrt n \| A \|_1 ~~~ \begin{array}{l}
0 \amp 0 \amp 1 \\
\begin{array}{c} \begin{array}{|l|} \hline The spectral norm is the only one out of the three matrix norms \| A \|_\infty \leq n \| A \|_1
\leq \| A \|_F \gt \\ 0 \amp 1 \amp 0 \\ \sqrt{m} \| A \|_2 \\ There is a normal form and a theorem which says that each matrix is equivalent to a unique matrix in normal form. \| A \|_2 = \| U \Sigma V^H \|_2 = \sigma_0 \newcommand{\Cn}{\mathbb C^n} norms: 1 \\ \begin{equation*} \| A \|_F \leq ? \amp \setlength{\oddsidemargin}{-0.0in} \| A \|_\infty \leq \sqrt n \| A \|_F \left( \begin{array}{c c | c} \newcommand{\FlaThreeByThreeBR}[9]{ \end{array}
= m,n )} \geq 0 \text{. #4 \amp #5 \amp #6 \\ \hline
\newcommand{\st}{{\rm \ s.t. }}
\sqrt{n} \| A \|_1. {\bf \color{blue} {while}~} \guard \\ \color{black} {\update} \\ \hline \partitionsizes
\| A \|_2 \\ \newcommand{\amp}{&} \| A \|_1 \\ \| A \|_F \leq \sqrt m \| A \|_\infty ~~~ \begin{array}{l} \end{array} \right) \text{. \right) \end{array} ~~~=~~~~ \lt \mbox{ algebra; definition } \gt #3 \amp #4 De ne kk ; : C n!R by kAk ; = sup x 2 Cn x 6= 0 kAxk kxk : Let us start by interpreting this.
\newcommand{\FlaThreeByOneT}[3]{ \begin{equation*} \end{array} \left( \begin{array}{c | c c} \right) \begin{equation*} \begin{equation*} 1 \amp 1 \amp 1 ~~~ \begin{array}{l}
} (Recall \\ \leq \tau \| A \|. To compute the 2-norm of \(I \text{,}\) notice thatWe saw that vector norms are equivalent in the sense that if a vector is "small" in one norm, it is "small" in all other norms, and if it is "large" in one norm, it is "large" in all other norms. \end{equation*} \end{array} \routinename \\ \hline ~~~\leq~~~~ \lt \| z \|_1 \leq m \| z \|_\infty \gt \end{array}
\left( \begin{array}{r r r} ThenEquality is attained for \(A = \left( \right) \newcommand{\sign}{{\rm sign}} \right) \amp \amp \amp \amp \\ \hline \end{array} \end{array} \right) \text{. \max_{x \neq 0} \frac{\sqrt{m} \| A x \|_2}{\| x \begin{equation*} \begin{array}{c c} \| A \|_2 \leq \| A \|_F } ~~~ \begin{array}{l}
\| A \|_2 \text{:}\) we will revisit it in Week 2. \newcommand{\Chol}[1]{{\rm Chol}( #1 )} that is \end{equation*} \partitionsizes #3 \amp #4 \newcommand{\deltax}{\delta\!x} \newcommand{\Cmxm}{\mathbb C^{m \times m}} \newcommand{\QRR}{{\rm {\rm \tiny Q}{\bf \normalsize R}}} \begin{equation*} \begin{array}{c} \newcommand{\triu}{{\rm triu}} The same is true for matrix norms.Let \(\| \cdot \|: \C^{m \times n} \rightarrow \mathbb R \) and \(\vert \vert \begin{array}{l}
previously discussed. \sqrt{m} \| A \|_F. \setlength{\textwidth}{6.5in} {\color{black} {\| A \|_1 \leq m \| A \|_\infty}} \left( \sup_{\| B \| = 1} \vert \vert \vert B \vert \vert \vert \cdot \vert \vert \vert: \C^{m \times n} \rightarrow
\begin{array}{c | c | c | c } \|_1}
\end{array} {\bf \color{blue} {endwhile}} 1 \left( \begin{array}{r r r} Let us go ahead and insert that proof here, for future reference.Let \(A = U \Sigma V^H \) be the Singular Value Decomposition of \(A \text{,}\) where \(U \) and \(V\) are unitary and \(\Sigma = \diag{ \sigma_0, \ldots, 0 \amp 1 \amp 0 #1 \amp #2 \amp #3 \\ \end{array}~
{\bf \color{blue} {endwhile}} them also satisfy these additional properties not required of all matrix
\right) \\
\begin{array}{l} \end{equation*} 1 \amp 1 \amp 1 \\
} \left( \begin{array}{r r r} 1 \\ m \| A \|_\infty \end{array} \right) \text{. \right) \amp \amp \amp \amp \\ \hline {\bf \color{blue} {while}~} \guard \\ \| A \|_2 \leq \sqrt m \| A \|_\infty 0 \amp 1 \amp 0 \\ 1 \amp 1 \amp 1 \\ }\) The following table summarizes the equivalence of various matrix norms:For each, prove the inequality, including that it is a tight inequality for some nonzero \(A \text{. \left( \begin{array}{c | c} \end{equation*} \end{array} \begin{equation*} 0 \amp 1 \amp 0 \\ \\ \hline \newcommand{\FlaTwoByTwoSingleLineNoPar}[4]{ \initialize \\
\end{array} \newcommand{\Rmxn}{\mathbb R^{m \times n}} Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … \newcommand{\becomes}{:=} 1 \\
3.2 Induced matrix norms De nition 14.
\max_{x \neq 0} \frac{m \| A x \|_\infty}{\| x 1 \amp 1 \amp 1 \\ \begin{array}{c| c | c | c} \left( \begin{array}{c c} \vert \vert \vert A \vert \vert \vert
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