Short Answer

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

$\lim_{t \to \infty} R^{t} = 0$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 $|R_{ii}| < 1 for i = 1, . . ., n$ .Now consider $λ$ , for

$to be the maximum value of all |R_{ii}|, for i = 1, 2, . . ., n$

Therefore $|λ| < 1$

$|R| \leq λ (l_{re} + U)$

is an upper triangular matrix such that $u_{ii} = 0 and u_{ij} = \frac{|r_{ij}|}{λ}, if j > i$ , if we have

$U^{n} = 0 Now consider |R^{t}| \leq {|R|}^{t} \leq λ^{t} {(l_{n} + U)}^{l} (As |R| \leq λ (l_{n} + U)) \leq λ^{i} t^{n} (l_{n} + U + U^{2} + L + U^{n - 1}) Also as |λ| < 1, we get from calculus that \lim_{t \to \infty} λ^{t} t^{n} = 0 Then \lim_{t \to \infty} R^{t} = 0$

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Linear Algebra With Applications

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Step 1: Define Upper Triangular Matrix:

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

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Linear Algebra With Applications

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Step 1: Define Upper Triangular Matrix:

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

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