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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