88 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRA
Proof: Let v1 be a unit eigenvector for A . Then there exists λ 1 such that
Av1 = λ 1v1, |v1|= 1.
Extend {v1} to a basis and then use Lemma 5.8.1 to obtain {v1, · · · ,vn}, an orthonormalbasis in Fn. Let U0 be a matrix whose ith column is vi. Then from the above, it follows U0is unitary. Then U∗0 AU0 is of the form
λ 1 ∗ · · · ∗0... A10
where A1 is an n− 1× n− 1 matrix. Repeat the process for the matrix, A1 above. Thereexists a unitary matrix Ũ1 such that Ũ∗1 A1 Ũ1 is of the form
λ 2 ∗ · · · ∗0... A20
.
Now let U1 be the n×n matrix of the form(1 00 Ũ1
).
This is also a unitary matrix because by block multiplication,(1 00 Ũ1
)∗( 1 00 Ũ1
)=
(1 00 Ũ∗1
)(1 00 Ũ1
)=
(1 00 Ũ∗1 Ũ1
)=
(1 00 I
)Then using block multiplication, U∗1 U∗0 AU0U1 is of the form
λ 1 ∗ ∗ · · · ∗0 λ 2 ∗ · · · ∗0 0...
... A20 0
where A2 is an n−2×n−2 matrix. Continuing in this way, there exists a unitary matrix,U given as the product of the Ui in the above construction such that
U∗AU = T
where T is some upper triangular matrix. Since the matrix is upper triangular, the charac-teristic equation is ∏
ni=1 (λ −λ i) where the λ i are the diagonal entries of T. Therefore, the
λ i are the eigenvalues.What if A is a real matrix and you only want to consider real unitary matrices?