这是一份uwa西澳大学MATH3022的成功案例

科学和工业建模|MATH3022 Scientific and Industrial Modelling代写 UWA代写

Similarly, there are many matrix norms. In particular, for each vector norm $|\cdot|$, we define a matrix norm by
$$
|A|=\max {\boldsymbol{x} \neq 0} \frac{|\boldsymbol{A} x|}{|x|} . $$ Computing a matrix norm using this definition is a difficult problem, but fortunately there are shortcuts. It can be shown that for an $m \times n$ matrix $\boldsymbol{A}$ $$ \begin{aligned} |\boldsymbol{A}|{1} &=\max {j} \sum{i=1}^{m}\left|a_{i j}\right|, \
|\boldsymbol{A}|_{2} &=\sqrt{\max {j} \lambda{j}\left(\boldsymbol{A}^{} \boldsymbol{A}\right),} \ |\boldsymbol{A}|_{\infty} &=\max {i} \sum{j=1}^{n}\left|a_{i j}\right|,
\end{aligned}
$$
where $\lambda_{j}\left(\boldsymbol{A}^{} \boldsymbol{A}\right)(j=1, \ldots, n)$ denotes the eigenvalues of the conjugate transpose of $\boldsymbol{A}$ times $\boldsymbol{A}$. One other norm, the Frobenius norm, is also useful. It is the 2 -norm of the matrix $\boldsymbol{A}$ after stacking its columns:
$$
|\boldsymbol{A}|_{F}=\sqrt{\sum_{j=1}^{n} \sum_{i=1}^{m}\left|a_{i j}\right|^{2}} .
$$

MATH3022 COURSE NOTES ：

We might want to know how close the solution to this problem is to our original problem. To answer this question, define
$$
\begin{aligned}
\epsilon_{A} & \equiv \frac{|\Delta A|}{|A|}, \
\epsilon_{b} & \equiv \frac{|\Delta b|}{|b|} \
\kappa & \equiv|A|\left|A^{-1}\right|
\end{aligned}
$$
Then if $\kappa \epsilon_{A}<1$ we have the bound
$$
\frac{|x-y|}{|x|} \leq \frac{\kappa}{1-\kappa \epsilon_{A}}\left(\epsilon_{A}+\epsilon_{b}\right)
$$
Consult a standard text such as $[64,119,139]$ for a proof of this and related results, valid for any of the norms $(1,2, \infty)$ we have considered. We already used a special case of this inequality, when $\epsilon_{b}=0$.