Linear Algebra, Theory And Applications, 2012a

7.4. SCHUR’S THEOREM 177
and furthermore, if the eigenvalues of A are listed in decreasing order,
λ 1 ≥ λ 2 ≥···≥λ n
Q can be chosen such that T is of the form
⎛
λ 1 ∗ ··· ∗
⎞
0 ··· 0 λ n . 0 λ .. . 2 ⎜
⎝
.
. .. . ⎟ .. ∗ ⎠
Proof: Most of this follows right away from Theorem 7.4.6. It remains to verify the
claim that the diagonal entries can be arranged in the desired order. However, this follows
from a simple modification of the above argument. When you find v 1 the eigenvalue of λ 1 ,
just be sure λ 1 is chosen to be the largest eigenvalue. Then observe that from Lemma 7.4.5
applied to the characteristic equation, the eigenvalues of the (n − 1) × (n − 1) matrix A 1
are {λ 1 , ··· ,λ n }.Thenpickλ 2 to continue the process of construction with A 1 .
Of course there is a similar conclusion which can be proved exactly the same way in the
case where A has complex eigenvalues.
Corollary 7.4.8 Let A be a real n × n matrix. Then there exists a real orthogonal matrix
Q and an upper triangular matrix T such that
⎛
⎞
P 1 ··· ∗
Q T ⎜
AQ = T = ⎝
. .. .
⎟
⎠
0 P r
where P i equals either a real 1 × 1 matrix or P i equals a real 2 × 2 matrix having as its
eigenvalues a conjugate pair of eigenvalues of A. If P k corresponds to the two eigenvalues
α k ± iβ k ≡ σ (P k ) ,Qcan be chosen such that
where
|σ (P 1 )|≥|σ (P 2 )|≥···
|σ (P k )|≡
√
α 2 k + β2 k
The blocks, P k can be arranged in any other order also.
Definition 7.4.9 When a linear transformation, A, mapping a linear space, V to V has a
basis of eigenvectors, the linear transformation is called non defective. Otherwise it is called
defective. An n × n matrix A, is called normal if AA ∗ = A ∗ A. An important class of normal
matrices is that of the Hermitian or self adjoint matrices. An n × n matrix A is self adjoint
or Hermitian if A = A ∗ .
The next lemma is the basis for concluding that every normal matrix is unitarily similar
to a diagonal matrix.
Lemma 7.4.10 If T is upper triangular and normal, then T is a diagonal matrix.
Proof:This is obviously true if T is 1 × 1. In fact, it can’t help being diagonal in this
case. Suppose then that the lemma is true for (n − 1) × (n − 1) matrices and let T be an
upper triangular normal n × n matrix. Thus T is of the form
( ) ( )
t11 a
T =
∗
,T ∗ t11 0
=
T
0 T 1 a T1
∗