Q49E Let R聽be a complex upper triang... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q49E (page 394)

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

Short Answer

Expert verified

$\lim_{t 鈫� 鈭�} R^{t} = 0$

Step by step solution

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

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 $位$ , for

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

Therefore $|位| < 1$

$|R| 鈮� 位 (l_{re} + U)$

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

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

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91影视

Short Answer

Step by step solution

Define Upper Triangular Matrix:

Upper triangular matrix with modulus of the entry:

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