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Chapter 7: Q.7.1-65E (page 325)

Consider an n 脳 n matrix A. A subspace V of Rn is said to be A invariant if Av鈫�is in V for all v鈫�in V. Describe all the one-dimensional A-invariant subspaces of Rn , in terms of the eigenvectors of A.

Short Answer

Expert verified

The solution is Av鈫�=位v鈫�

Step by step solution

01

V is one dimensional A-invariant:

Let V be one dimensional A- invariant substance of Rn, and v鈫� a nonzero vector or in V.

Then Av鈫�will be in V, so that Av鈫�=位v鈫�for some 位

v鈫� is an eigenvector of A.

02

v→ is any eigenvector of A:

If is any eigenvector of A, then

V=span(v鈫�) will be a one dimensional A-invariant substance.

Then one dimensional A-invariant substance V are of the form V=span(v鈫�),

Where v鈫�is an eigenvector of A.


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