Page 62 - Matrices theory and applications

3.2. Spectral Decomposition of Normal Matrices
45
Corollary 3.1.1 In M n(k) (k = IR or CC) the set of diagonalizable
matrices is an open subset.
3.1.2 Trigonalization in an Orthonormal Basis
From now on we say that two matrices are unitarily similar if they are
similar through a unitary transformation. Two real matrices are unitarily
similar if they are similar through an orthogonal transformation.
If K = CC, one may sharpen Theorem 2.7.1:
Theorem 3.1.3 (Schur) If M ∈ M n (CC), there exists a unitary matrix
∗
U such that U MU is upper triangular.
One also says that every matrix with complex entries is unitarily
trigonalizable.
Proof
We proceed by induction on the size n of the matrices. The statement is
trivial if n = 1. Let us assume that it is true in M n−1 (CC), with n ≥ 2. Let
M ∈ M n (CC) be a matrix. Since CC is algebraically closed, M has at least
one eigenvalue λ.Let X be an eigenvector associated to λ. By dividing X
by X , one can assume that X is a unit vector. One can then find an
2
1
n
n
orthonormal basis {X ,X ,... ,X } of CC whose first element is X.Let
2
n
1
us consider the matrix V := (X = X, X ,... ,X ), which is unitary, and
let us form the matrix M := V MV .Since

∗
1
1
1
VM e = MV e = MX = λX = λV e ,
1
1

one obtains M e = λe .In other words, M has the block-triangular form:
λ ···

M = ,
0 n−1 N
where N ∈ M n−1 (CC). Applying the induction hypothesis, there exists
ˆ
∗
W ∈ U n−1 such that W NW is upper triangular. Let us denote by W
ˆ
ˆ ∗
the (block-diagonal) matrix diag(1,W) ∈ U n .Then W M W is upper
ˆ
triangular. Hence, U = V W satisfies the conditions of the theorem.
3.2 Spectral Decomposition of Normal Matrices
We recall that a matrix M is normal if M commutes with M. For real
∗
matrices, this amounts to saying that M T commutes with M. Since it is
equivalent for a Hermitian matrix H to be zero or to satisfy x Hx =0 for
∗
∗
every vector x,we see that M is normal if and only if Ax 2 = A x 2
for every vector, where x 2 denotes the standard Hermitian (Euclidean)
∗
norm (take H = AA − A A).
∗