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EXAMPLE 1 An Orthogonal Matrix P That Diagonalizes a Matrix A
Find an orthogonal matrix P that diagonalizes
Solution
The characteristic equation of A is
Thus the eigenvalues of A are and . By the method used in Example 5 of Section 7.1, it can be shown that
(5)
form a basis for the eigenspace corresponding to . Applying the Gram–Schmidt process to yields the
following orthonormal eigenvectors (verify):
(6)
The eigenspace corresponding to has
as a basis. Applying the Gram–Schmidt process to yields
Finally, using , , and as column vectors, we obtain
which orthogonally diagonalizes A. (As a check, the reader may wish to verify that is a diagonal matrix.)
