Short Answer

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

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

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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 zeros on the diagonal:

Consider as $U \leq 0$ be a real upper triangular matrix $n \times n$ with zeros on the diagonal. Therefore is U a nilpotent

$U^{n} = 0$

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

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

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

${(l_{n} + U)}^{t} \leq t^{n} (l_{n} + U + U^{2} + . . . . . + U^{n - 1})$

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Define Upper Triangular Matrix:

Step 2: Upper triangular matrix with zeros on the diagonal:

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Linear Algebra With Applications

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Short Answer

Let U⩾0 be a real upper triangular n×n matrix with zeros on the diagonal. Show that (ln+U)t⩽tn(ln+U+U2+.....+Un-1)for all positive integers t. See Exercises 46 and 47.

Step by Step Solution

Step 1: Define Upper Triangular Matrix:

Step 2: Upper triangular matrix with zeros on the diagonal:

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