Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Linear Algebra With Applications
Found in: Page 394
Linear Algebra With Applications

Linear Algebra With Applications

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

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Let R be a complex upper triangular nxn matrix with |rii|< for i,...,n . Show that

limtRt=0 ,,

meaning that the modulus of all entries of Rt approaches zero. Hint: We can write |R|(ln+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.


See the step by step solution

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 Rii<1 for i=1,...,n.Now consider λ, for

to be the maximum value of all Rii, for i=1,2,...,n

Therefore λ<1


is an upper triangular matrix such that uii=0 and uij=rijλ,if j>i, if we have

Un=0Now considerRtRtλtln+Ul AsRλln+Uλitnln+U+U2+L+Un-1 Also as λ<1,we get from calculus thatlimtλttn=0Then limtRt=0

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.