Properties of Eigenvalues and Eigenvectors:



Properties of Eigenvalues and Eigenvectors:

(a)

Let [pic] be the eigenvector of [pic] associated with the eigenvalue [pic]. Then, the eigenvalue of

,

associated with the eigenvector [pic] is

[pic],

where [pic] are real numbers and [pic] is a positive integer.

[proof:]

[pic]

since

[pic].

Example:

[pic],

what is the eigenvalues of [pic].

[solution:]

The eigenvalues of A are -5 and 7. Thus, the eigenvalues of [pic] are

[pic]

and

[pic].

Example:

Let [pic] be the eigenvalue of A. Then, we denote

[pic].

Then, [pic] has eigenvalue

[pic].

Note:

Let [pic] be the eigenvector of A associated with the eigenvalue [pic]. Then, [pic] is the eigenvector of [pic] associated with the eigenvalue [pic].

[proof:]

[pic].

Therefore, [pic] is the eigenvector of [pic] associated with the eigenvalue [pic].

(b)

Let [pic] be the eigenvalues of A ([pic] are not necessary to be distinct). Then,

[pic] and [pic].

[proof:]

[pic].

Thus,

[pic]

Therefore,

[pic].

Also, by diagonal expansion on the following determinant

[pic],

and by the expansion of

[pic],

therefore,

[pic].

Example:

[pic],

The eigenvalues of A are [pic] and [pic]. Then,

[pic]

and

[pic].

-----------------------

[pic]

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