Q48E Let聽U猢�0聽be a real upper tria... [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: Q48E (page 393)

Let $U 猢� 0$ be a real upper triangular $n 脳 n$ matrix with zeros on the diagonal. Show that
${(l_{n} + U)}^{t} 猢� t^{n} (l_{n} + U + U^{2} + . . . . . + U^{n - 1})$
for all positive integers t. See Exercises 46 and 47.

Short Answer

Expert verified

${(l_{n} + U)}^{t} {leqt}^{n} (l_{n} + U + U^{2} + . . . . . + U^{n - 1}) for all positive integers t$

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 zeros on the diagonal:

Consider as $U 鈮� 0$ be a real upper triangular matrix $n 脳 n$ with zeros on the diagonal. Therefore is U a nilpotent

$U^{n} = 0$

Now consider ${(l_{n} + U)}^{t}$ for $t 鈮� n - 1$ as represented below:

${(l_{n} + U)}^{t} = l_{n} + {鈭�}_{k = 1}^{t} (\begin{matrix} t \\ k \end{matrix}) U^{k} {(l_{n} + U)}^{t} = l_{n} + (\begin{matrix} t \\ 1 \end{matrix}) U + (\begin{matrix} t \\ 2 \end{matrix}) U^{2} + . . . . + (\begin{matrix} t \\ n - 1 \end{matrix}) U^{n - 1} (\begin{matrix} l \\ k \end{matrix}) 鈮� t^{k}, f o r k = 0, 1, . . . . ., n - 1 l_{n} + (\begin{matrix} t \\ 1 \end{matrix}) U + (\begin{matrix} t \\ 2 \end{matrix}) U^{2} + . . . . + (\begin{matrix} t \\ n - 1 \end{matrix}) U^{n - 1} 鈮� t^{n} (l_{n} + U + U^{2} + . . . . + U^{n - 1})$

substituting (1) in the above inequality we get as represented below:

${(l_{n} + U)}^{t} 鈮� t^{n} (l_{n} + U + U^{2} + . . . . . + U^{n - 1})$

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91影视

Let $U 猢� 0$ be a real upper triangular $n 脳 n$ matrix with zeros on the diagonal. Show that
${(l_{n} + U)}^{t} 猢� t^{n} (l_{n} + U + U^{2} + . . . . . + U^{n - 1})$
for all positive integers t. See Exercises 46 and 47.

Short Answer

Step by step solution

Define Upper Triangular Matrix:

Upper triangular matrix with zeros on the diagonal:

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

LetU猢�0be a real upper triangularn脳n matrix with zeros on the diagonal. Show that(ln+U)t猢�tn(ln+U+U2+.....+Un-1)for all positive integers t. See Exercises 46 and 47.

Short Answer

Step by step solution

Define Upper Triangular Matrix:

Upper triangular matrix with zeros on the diagonal:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Geometry

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Let $U 猢� 0$ be a real upper triangular $n 脳 n$ matrix with zeros on the diagonal. Show that
${(l_{n} + U)}^{t} 猢� t^{n} (l_{n} + U + U^{2} + . . . . . + U^{n - 1})$
for all positive integers t. See Exercises 46 and 47.