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Chapter 7: Q44E (page 358)

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.

T(x0,x1,x2,...)=T(x2,x3,...)From the spaceof infinite sequences into. (We drop the first two terms of the sequence.)

Short Answer

Expert verified

The eigenvalues and eigenvectors of the linear equation is,

位鈭�R,E位=span1,0,位,0,位2,0,...,0,1,0,位,0,位2,,...

Step by step solution

01

Define eigenvalues

The scalar values that are associated with the vectors of the linear equations in the matrix are called eigenvalues.

Ax鈫�=位x鈫�,here x鈫�is eigenvector and位 is the eigenvalue.

02

Calculate the eigenvalues and eigenvectors by substituting the given equation

Consider the given equation,

T(x0,x1,x2,...)=T(x2,x3,...)

Solve,

T(x0,x1,x2,...)=位(x0,x1,x2,...)(x2,x3,x4,...)=位(x0,x1,x2,...)xn=位xn-2,n鈮�2

Therefore, for an eigenvalue 位鈭�R, we have eigenvectors,

E位=span1,0,位,0,位2,0,...,0,1,0,位,0,位2,,...

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Most popular questions from this chapter

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 line L inR2.

Consider the matrix A=|abb-a|where aand bare arbitrary constants. Find all eigenvalues of A. Explain in terms of the geometric interpretation of the linear transformationT(x)鈫�=Ax鈫�.

IfAis a matrix of rank 1, show that any non-zero vector in the image of Ais an eigenvector of A.

Find a basis of the linear space Vof all3脳3matrices Afor which both[100]and[001]are eigenvectors, and thus determine the dimension of.

Find a basis of the linear space Vof all2脳2matrices Afor which[1-3] is an eigenvector, and thus determine the dimension of V.
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