Page 65 - Matrices theory and applications

3. Matrices with Real or Complex Entries
48
block-diagonal matrix whose diagonal blocks are direct similitudes. Hence
M 1 and M are unitarily similar to such a matrix.
Corollary 3.3.1 Real symmetric matrices are diagonalizable over IR,
through orthogonal conjugation. In other words, given M ∈ Sym (IR),
n
−1
is diagonal.
there exists an O ∈ O n (IR) such that OMO
In fact, since the eigenvalues of M are real, OMO −1 has only 1 × 1blocks.
We say that real symmetric matrices are orthogonally diagonalizable.
The interpretation of this statement in terms of quadratic forms is the
n
following. For every quadratic form Q on IR , there exists an orthonor-
mal basis {e 1,... ,e n} in which this form can be written with at most n
squares: 4
n
2
Q(x)= a i x .
i
i=1
1/2
Replacing the basis vector e j by |a j | e j , one sees that there also exists
an orthogonal basis in which the quadratic form can be written
r s
2 2
Q(x)= x − x ,
i j+r
i=1 j=1
with r+s ≤ n. This quadratic form is nondegenerate if and only if r+s = n.
The pair (r, s) is unique and called the signature or the Sylvester index of
the quadratic form. In such a basis, the matrix associated to Q is
1
 
.
 . 
 . 0 
 
1
 
 
−1
 
 
.
 .  .

 .
 
 −1 
 
 0 
 
.
 . 
 0 . 
0
3.3.1 Rayleigh Quotients
Let M be a real n × n symmetric matrix, and let λ 1 ≤ ··· ≤ λ n be its
eigenvalues arranged in increasing order and counted with multiplicity. Let
4
In solid mechanics, when Q is the matrix of inertia, the vectors of this basis are
along the inertia axes,and the a j , which then are positive, are the momenta of inertia.