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Chapter 7: Q7-22E (page 383)

TRUE OR FALSE

22. If vectorv鈫�is an eigenvector of both A and B, thenv鈫� must be an eigenvector of A+B.

Short Answer

Expert verified

The given statement is true.

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�, herex鈫�is the eigenvector and位is the eigenvalue.

02

Use the formula to find an eigenvector

Consider is an eigenvector of A and B, then,

Av=位v,Bv=渭v

Thus,

(A+B)v=Av+Bv=位v+渭v=(位+渭)v

Hence, v is also an eigenvector of A+B.

Therefore, the given statement is true.

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