Linear Algebra, Theory And Applications, 2012a

7.4. SCHUR’S THEOREM 179
Proof: From Theorem 7.4.4 there exists a unitary matrix U such that U ∗ AU equals
an upper triangular matrix. The theorem is now proved if it is shown that the property of
being normal is preserved under unitary similarity transformations. That is, verify that if
A is normal and if B = U ∗ AU, then B is also normal. But this is easy.
B ∗ B = U ∗ A ∗ UU ∗ AU = U ∗ A ∗ AU
= U ∗ AA ∗ U = U ∗ AUU ∗ A ∗ U = BB ∗ .
Therefore, U ∗ AU is a normal and upper triangular matrix and by Lemma 7.4.10 it must be
a diagonal matrix.
Corollary 7.4.12 If A is Hermitian, then all the eigenvalues of A are real and there exists
an orthonormal basis of eigenvectors.
Proof: Since A is normal, there exists unitary, U such that U ∗ AU = D, a diagonal
matrix whose diagonal entries are the eigenvalues of A. Therefore, D ∗ = U ∗ A ∗ U = U ∗ AU =
D showing D is real.
Finally, let
U = ( u 1 u 2 ··· u n
)
where the u i denote the columns of U and
⎛
⎞
λ 1 0
⎜
D = ⎝
. ..
⎟
⎠
0 λ n
The equation, U ∗ AU = D implies
AU = ( )
Au 1 Au 2 ··· Au n
= UD = ( )
λ 1 u 1 λ 2 u 2 ··· λ n u n
where the entries denote the columns of AU and UD respectively. Therefore, Au i = λ i u i
and since the matrix is unitary, the ij th entry of U ∗ U equals δ ij and so
δ ij = u ∗ i u j ≡ u j · u i .
This proves the corollary because it shows the vectors {u i } are orthonormal. Therefore,
they form a basis because every orthonormal set of vectors is linearly independent.
Corollary 7.4.13 If A is a real symmetric matrix, then A is Hermitian and there exists a
real unitary matrix U such that U T AU = D where D is a diagonal matrix whose diagonal
entries are the eigenvalues of A. By arranging the columns of U the diagonal entries of D
can be made to appear in any order.
Proof: This follows from Theorem 7.4.6 and Corollary 7.4.12. Let
U = ( u 1 ··· u n
)
Then AU = UD so
AU = ( Au 1 ··· Au n
) (
= u1 ··· u n
) (
D = λ1 u 1 ··· λ n u n
)
Hence each column of U is an eigenvector of A. It follows that by rearranging these columns,
the entries of D on the main diagonal can be made to appear in any order. To see this,
consider such a rearrangement resulting in an orthogonal matrix U ′ given by
U ′ = ( u i1 ··· u in
)