Chapter 8: Chapter 8

Problem 188

$\begin{array}{cccccccc} & /0 & 0 & 0 & 0 & 1 & 1 & 1 \mid \\ \text { Let } \mathrm{A}= & /0 & 2 & 6 & 2 & 0 & 0 & 4 \mid . \\ &/0 & 1 & 3 & 1 & 2 & 1 & 2 \mid\end{array}$ Reduce A to the Hermite normal form.

Short Answer

Expert verified

The Hermite normal form of matrix A is: $HNF(A) = \begin{bmatrix} 0 & 0 & 0 & 0 & 4 & 2 & 0 \\ 0 & 1 & 3 & 1 & -2 & -1 & 2 \end{bmatrix}$

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Ensure positive diagonal values

In the given matrix, the first diagonal element is already positive, so we don't need to make any changes: $A = \begin{bmatrix} 0 & 2 & 6 & 2 & 0 & 0 & 4 \\ 0 & 1 & 3 & 1 & 2 & 1 & 2 \end{bmatrix}$

Step 2: Create zeros below the first nonzero number in a nonzero row

To achieve this, we can perform the following row operation R2 = R2 - R1: $\begin{bmatrix} 0 & 2 & 6 & 2 & 0 & 0 & 4 \\ 0 & -1 & -3 & -1 & 2 & 1 & -2 \end{bmatrix}$

Step 3: Make the first nonzero number in a nonzero row positive

For the second row, R2 = -R2: $\begin{bmatrix} 0 & 2 & 6 & 2 & 0 & 0 & 4 \\ 0 & 1 & 3 & 1 & -2 & -1 & 2 \end{bmatrix}$

Step 4: Create zeros above the first nonzero number in a nonzero row

Perform the row operation R1 = R1 - 2 * R2: $\begin{bmatrix} 0 & 0 & 0 & 0 & 4 & 2 & 0 \\ 0 & 1 & 3 & 1 & -2 & -1 & 2 \end{bmatrix}$

Step 5: Ensure that the first nonzero number in a nonzero row divides all other numbers in the row

The current matrix meets this requirement: $\begin{bmatrix} 0 & 0 & 0 & 0 & 4 & 2 & 0 \\ 0 & 1 & 3 & 1 & -2 & -1 & 2 \end{bmatrix}$ Now we have the Hermite normal form of matrix A: $HNF(A) = \begin{bmatrix} 0 & 0 & 0 & 0 & 4 & 2 & 0 \\ 0 & 1 & 3 & 1 & -2 & -1 & 2 \end{bmatrix}$

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Chapter 8: Chapter 8

$\begin{array}{cccccccc} & /0 & 0 & 0 & 0 & 1 & 1 & 1 \mid \\ \text { Let } \mathrm{A}= & /0 & 2 & 6 & 2 & 0 & 0 & 4 \mid . \\ &/0 & 1 & 3 & 1 & 2 & 1 & 2 \mid\end{array}$ Reduce A to the Hermite normal form.

Short Answer

Step by step solution

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Step 1: Ensure positive diagonal values

Step 2: Create zeros below the first nonzero number in a nonzero row

Step 3: Make the first nonzero number in a nonzero row positive

Step 4: Create zeros above the first nonzero number in a nonzero row

Step 5: Ensure that the first nonzero number in a nonzero row divides all other numbers in the row

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More chapters from the book ‘Linear Algebra’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

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