12.5 Norms of Matrixes[2CN]

Let then \(p,q∈[1,∞]\); we use the following norms \(|x|_ p\) defined in eqn. ??.

Definition 339

[11G] Let \(A∈ℝ^{m× n}\) be a matrix; considering it as a linear application between normed spaces \((ℝ^ n,||_ p)\) and \((ℝ^ m,||_ q)\), let’s define again the induced norm as

\begin{equation} \| A\| _{p,q}{\stackrel{.}{=}}\max _{x∈ℝ^ n~ ,~ |x|_ p≤ 1} |Ax|_ q\label{eq:norme_ matrici} \end{equation}
340

(Note that the maximum is always reached at a point with \(|x|_ p=1\)).

The norm \(\| A\| _{2,2}\) is called the spectral norm. .

Definition 341

[11H] We also define the rules

\[ \| A\| _{F-\tilde p}= \begin{cases} \sqrt[\tilde p]{∑_{i,j} |A_{i,j}|^{\tilde p}} & \tilde p {\lt}∞~ ~ ~ ,\\ \max _{i,j} |A_{i,j}| & \tilde p =∞ \end{cases} \]

for \(\tilde p∈[1,∞]\). The case \(\tilde p=2\) is called Frobenious’ norm.

E343

[11J] Prerequisites:325.Note that the norms \(\| A\| _{p,q}\) and \(\| A\| _{F-\tilde p}\) are all equivalent.

E343

[11K] Prerequisites:4.Let’s consider square matrices, i.e. \(n=m\). We know from 4 that norms \(\| A\| _{p,q}\) are submultiplicative, that is \(\| A B\| _{p,q}≤ \| A\| _{p,q} \| A\| _{p,q} \).

Show that the Frobenious norm is also submultiplicative.

Note that for a submultiplicative norm we have that \(\| A^ k\| ≤ \| A\| ^ k\) for every natural \(k\).

E343

[11M] Show that

\[ \left\| A \right\| _{1,1} = \max _{1 ≤ j ≤ n} ∑ _{i=1} ^ m | A_{i,j} |, \]
\[ \left\| A \right\| _{∞,∞} = \max _{1 ≤ i ≤ m} ∑ _{j=1} ^ n | A_{i,j} |~ ~ . \]
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[11N]If \(A∈ℂ^{m× n}\) we can define the induced norms

\begin{equation} \| A\| _{p,q}{\stackrel{.}{=}}\max _{x∈ℂ^ n~ ,~ |x|_ p≤ 1} |Ax|_ q~ ~ .\label{eq:norme_ matrici_ C} \end{equation}
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Show that \(\| A\| _{p,q} = \| \overline A\| _{p,q}\).

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[11P]Show that if \(A∈ℝ^{m× n}\) you have

\[ \max _{x∈ℝ^ n, |x|_ 2≤ 1} |Ax|_ 2=\max _{x∈ℂ^ n, |x|_ 2≤ 1} |Ax|_ 2~ ~ . \]

Hidden solution: [UNACCESSIBLE UUID ’11Q’]