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Theorem 10.6.1 to write (with the Euclidean inner product on ), then we obtain
But if we agree not to distinguish between the matrix and its entry, and if we use the fact
that eigenvectors are nonzero, then we can express as
(6)
Thus, to show that is a real number, it suffices to show that the entry of is real. One way to do
this is to show that the matrix is Hermitian, since we know that Hermitian matrices have real
numbers on the main diagonal. However,
which shows that is Hermitian and completes the proof.
The proof of the following theorem is an immediate consequence of Theorem 10.6.5 and is left as an exercise.
THEOREM 10.6.6
The eigenvalues of a symmetric matrix with real entries are real numbers.
Exercise Set 10.6
Click here for Just Ask!
In each part, find .
1.
(a)
(b)
(c)
(d)
