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Chapter 7: Q36E (page 338)

For an arbitrary positive integer n, give a 2n脳2n matrix A without real eigenvalues.

Short Answer

Expert verified

An Arbitrary positive integer.

Give a 2n脳2nmatrix A without real eigenvalues.

A=i00i

Step by step solution

01

definition of arbitrary positive integer

Any arbitrary positive integer n can be uniquely represented as the product of a powerful number.

An arbitrary positive integer n,

Give a 2n脳2nmatrix A without real eigenvalues.

For the example

A=i00i

Its characteristic polynomial is位-i2

02

definition of eigenvalues

The eigenvalue is a number that indicates how much variance exists in the data in that direction; in the example above, the eigenvalue is a number that indicates how spread out the data is on the line.

so, its only eigenvalue is i,

Then with algebraic multiplicity.

A=i00i

Hence,

A=i00i

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Most popular questions from this chapter

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[1234]

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[-3040-10-273]

If a vector v鈬赌is an eigenvector of both Aand B, isv鈬赌necessarily an eigenvector of A+B?

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise1 through20,find all (real)eigenvalues.Then find a basis of each eigenspaces,and diagonalize A, if you can. Do not use technology.

[1111]

find an eigenbasis for the given matrice and diagonalize:

A=114[123246369]
See all solutions

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