Page 66 - Matrices theory and applications

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3.3. Normal and Symmetric Real-Valued Matrices
us denote by B = {v 1 ,... ,v n } an orthonormal eigenbasis (Mv j = λ j v j ).
n
If x ∈ IR , let us denote by y 1 ,... ,y n the coordinates of x in the basis B.
n
Finally, let us denote by · 2 the usual Euclidean norm on IR .Then

2
T
2
2
x Mx =
y = λ n x .
λ j y ≤ λ n
2
j
j
j
j
2
T
Since v Mv n = λ n v n , we deduce the value of the largest eigenvalue of
n
2
M:
T
x Mx T 2
λ n =max 2 =max x Mx | x =1 . (3.1)
2
x =0 x
2
Similarly, the smallest eigenvalue of a real symmetric matrix is given by
T
x Mx T 2
λ 1 =min 2 =min{x Mx | x =1}. (3.2)
2
x =0 x
2
For a Hermitian matrix, the formulas (3.1,3.2) remain valid when we replace
T
x by x .
∗
We evaluate the other eigenvalues of M ∈ Sym (IR) in the following
n
n
way. For every linear subspace F of IR of dimension k, let us define
T
x Mx T 2
R(F)= max =max x Mx | x ∈ F, x =1 .
2
x∈F \{0} x 2
2
The intersection of F with the linear subspace spanned by {v k ,... ,v n } is
of dimension greater than or equal to one. There exists, therefore, a nonzero
vector x ∈ F such that y 1 = ··· = y k−1 = 0. One has then
n
2
2
T
x Mx = 2 y = λ k x .
λ j y ≥ λ k
j j 2
j=k j
Hence, R(F) ≥ λ k .Furthermore, if G is the space spanned by {v 1 ,... ,v k },
one has R(G)= λ k .Thus, we have
λ k =min{R(F) | dim F = k}.
Finally, we may state the following theorem.
Theorem 3.3.2 Let M be an n × n real symmetric matrix and λ 1 ,... ,λ n
its eigenvalues arranged in increasing order, counted with multiplicity. Then
T
x Mx
λ k = min max 2 .
dim F =k x∈F \{0} x
2
If M is complex Hermitian, one has similarly
∗
x Mx
λ k = min max 2 .
dim F =k x∈F \{0} x
2
This formula generalizes (3.1, 3.2).