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Found in: Page 394

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Let R be a complex upper triangular nxn matrix with $|{r}_{\mathrm{ii}}|{\mathbf{<}}{\mathbf{}}{\mathbit{f}}{\mathbit{o}}{\mathbit{r}}{\mathbf{}}{\mathbit{i}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{\mathbit{n}}{\mathbf{}}$. Show that $\underset{\mathbf{t}\mathbf{\to }\mathbf{\infty }}{\mathbf{l}\mathbf{i}\mathbf{m}}{{\mathbit{R}}}^{{\mathbf{t}}}{\mathbf{=}}{\mathbf{0}}{\mathbf{}}{\mathbf{,}}$,meaning that the modulus of all entries of ${{\mathbit{R}}}^{{\mathbf{t}}}$ approaches zero. Hint: We can write $|R|{\mathbf{\le }}\left({l}_{n}+U\right)$, for some positive real number $|\lambda |{\mathbf{<}}{\mathbf{1}}$ and an upper triangular U > 0 matrix with zeros on the diagonal. Exercises 47 and 48 are helpful.

$\underset{\mathrm{t}\to \infty }{\mathrm{lim}}{R}^{t}=0$

See the step by step solution

## Step 1: Define Upper Triangular Matrix:

A triangular matrix with all components equal to below the main diagonal is called an upper triangular matrix. It's an element-based square matrix

## Step 2: Upper triangular matrix with modulus of the entry:

Let R be a upper triangular matrix nxn with $\left|{\mathrm{R}}_{\mathrm{ii}}\right|<1\mathrm{for}\mathrm{i}=1,...,\mathrm{n}$.Now consider $\lambda$, for

$\mathrm{to}\mathrm{be}\mathrm{the}\mathrm{maximum}\mathrm{value}\mathrm{of}\mathrm{all}\left|{\mathrm{R}}_{\mathrm{ii}}\right|,\mathrm{for}\mathrm{i}=1,2,...,\mathrm{n}$

Therefore $\left|\mathrm{\lambda }\right|<1$

$\left|\mathrm{R}\right|\le \lambda \left({\mathrm{l}}_{\mathrm{re}}+\mathrm{U}\right)$

is an upper triangular matrix such that ${\mathrm{u}}_{\mathrm{ii}}=0\mathrm{and}{\mathrm{u}}_{\mathrm{ij}}=\frac{\left|{\mathrm{r}}_{\mathrm{ij}}\right|}{\mathrm{\lambda }},\mathrm{if}\mathrm{j}>\mathrm{i}$, if we have

${\mathrm{U}}^{\mathrm{n}}=0\phantom{\rule{0ex}{0ex}}\mathrm{Now}\mathrm{consider}\phantom{\rule{0ex}{0ex}}\left|{\mathrm{R}}^{\mathrm{t}}\right|\le {\left|\mathrm{R}\right|}^{\mathrm{t}}\phantom{\rule{0ex}{0ex}}\le {\mathrm{\lambda }}^{\mathrm{t}}{\left({\mathrm{l}}_{\mathrm{n}}+\mathrm{U}\right)}^{\mathrm{l}}\left(\mathrm{As}\left|\mathrm{R}\right|\le \mathrm{\lambda }\left({\mathrm{l}}_{\mathrm{n}}+\mathrm{U}\right)\right)\phantom{\rule{0ex}{0ex}}\le {\mathrm{\lambda }}^{\mathrm{i}}{\mathrm{t}}^{\mathrm{n}}\left({\mathrm{l}}_{\mathrm{n}}+\mathrm{U}+{\mathrm{U}}^{2}+\mathrm{L}+{\mathrm{U}}^{\mathrm{n}-1}\right)\phantom{\rule{0ex}{0ex}}\mathrm{Also}\mathrm{as}\left|\mathrm{\lambda }\right|<1,\mathrm{we}\mathrm{get}\mathrm{from}\mathrm{calculus}\mathrm{that}\phantom{\rule{0ex}{0ex}}\underset{\mathrm{t}\to \infty }{\mathrm{lim}}{\mathrm{\lambda }}^{\mathrm{t}}{\mathrm{t}}^{\mathrm{n}}=0\phantom{\rule{0ex}{0ex}}\mathrm{Then}\underset{\mathrm{t}\to \infty }{\mathrm{lim}}{\mathrm{R}}^{\mathrm{t}}=0$