Page 64 - Matrices theory and applications

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3.3. Normal and Symmetric Real-Valued Matrices
3.3 Normal and Symmetric Real-Valued Matrices
The situation is a bit more involved if M, a normal matrix, has real en-
tries. Of course, one can consider M as a matrix with complex entries and
diagonalize it in an orthonormal basis, but we quit in general the field of
real numbers when doing so. We prefer to allow bases consisting of only
real vectors. Since some of the eigenvalues might be nonreal, one cannot in
general diagonalize M. The statement is thus the following.
Theorem 3.3.1 Let M ∈ M n (IR) be a normal matrix. There exists an or-
thogonal matrix O such that OMO −1 be block-diagonal, the diagonal blocks
being 1 × 1 (those corresponding to the real eigenvalues of M)or 2 × 2,the
latter being matrices of direct similitude: 3

a b
(b =0).
−ba
T
Similarly, OM O −1 is block-diagonal, the diagonal blocks being eigen-
values or matrices of direct similitude.
Proof
One again proceeds by induction on n.When n ≥ 1, the proof is the same
as in the previous section whenever M has at least one real eigenvalue.
If this is not the case, then n is even. Let us first consider the case n =2.
Then

a b
M = .
c d
2
2
Since M is normal, we have b = c and (a − d)(b − c)=0. However,
b = c, since otherwise M would be symmetric, hence would have two real
eigenvalues. Hence b = −c and a = d.
Now let us consider the general case, with n ≥ 4. We know that M has
an eigenpair (λ, z), where λ is not real. If the real and imaginary parts of
z were colinear, M would have a real eigenvector, hence a real eigenvalue,
a contradiction. In other words, the real and imaginary parts of z span a
¯
n
T
plane P in IR . As before, Mz = λz implies M z = λz. Hence we have
T
MP ⊂ P and M P ⊂ P.Now let V be an orthogonal matrix that maps
T
1
2
the plane P 0 := IRe ⊕ IRe onto P. Then the matrix M 1 := V MV is
normal and satisfies
T
M 1P 0 ⊂ P 0 , M P 0 ⊂ P 0 .
1
This means that M 1 is block-diagonal. Of course, each diagonal block (of
sizes 2×2and (n−2)×(n−2)) inherits the normality of M 1. Applying the
induction hypothesis, we know that these blocks are unitarily similar to a
3
A similitude is an endomorphism of a Euclidean space that preserves angles. It splits
as aR,where R is orthogonal and a is a scalar. It is direct if its determinant is positive.