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Chapter 7: Q7-12E (page 383)

TRUE OR FALSE

12. If a real matrix A has only the eigenvalues 1 and 鈭�1, then A must be orthogonal.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Define eigenvalue

Eigenvalues are a special set of scale values associated with a set of line numbers in almost matrix measurements.

The elements of the matrix, also known as the entries, are the integers. In addition to many other areas of mathematics, matrices have extensive applications in the fields of engineering, physics, economics, and statistics.

02

Explanation for matrix A has only the eigenvalues 1 and −1

For example, consider the matrix

A=-1101

This matrix is upper-triangular, so its eigenvalues are the entries on its main diagonal, which are 位1=-1,位2=1. However, its second column is not a unit vector, so A is not orthogonal.

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Most popular questions from this chapter

If v鈬赌 is an eigenvector of matrix A with associated eigenvalue 3 , show that v鈬赌is an image of matrix A .

Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.

Reflection about a plane v inR3.

If a 2 脳 2 matrix A has two distinct eigenvalues位1and位2, show that A is diagonalizable.

Arguing geometrically, find all eigen vectors and eigen values of the linear transformations. In each case, find an eigen basis if you can, and thus determine whether the given transformation is diagonalizable.

Scaling by 5 inR3.

find an eigenbasis for the given matrice and diagonalize:

A=19[82-2255-244]

Representing the orthogonal projection onto the planex-2y+2z=0
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