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212 ALGEBRA
n n

where P i = |a ij |,and P i can be replaced by Q i = |a ji |.
j=1, j≠i j=1, i≠i
The modulus of any eigenvalue λ of a Hermitian operator A in an n-dimensional unitary
space satisfies the inequalities

2
2
|λ| ≤ |a ij | , |λ| ≤ A =sup [(Ax) ⋅ x],
x =1
i j
and its smallest and its largest eigenvalues, denoted, respectively, by m and M, can be
found from the relations
m =inf [(Ax) ⋅ x], M =sup [(Ax) ⋅ x].
x =1 x =1

5.6.3-5. Spectral decomposition of Hermitian operators.

Let i 1 , ... , i n be a fixed orthonormal basis in an n-dimensional unitary space V.Then any
element of V can be represented in the form (see Paragraph 5.4.2-2)
n

x = (x ⋅ i j )i j .
j=1
The operator P k (k = 1, 2, ... , n)defined by

P k x =(x ⋅ i k )i k
is called the projection onto the one-dimensional subspace generated by the vector i k .The
projection P k is a Hermitian operator.
Properties of the projection P k :

for k = l,
P k P l = P k P m = P k (m = 1, 2, 3, ...),
O for k ≠ l, k
n

P j = I, where I is the identity operator.
j=1
For a normal operator A, there is an orthonormal basis consisting of its eigenvectors,
Ai k = λi k . Then one obtains the spectral decomposition of a normal operator:
n
k
k
A = λ P j (k = 1, 2, 3, ...). (5.6.3.3)
j
j=1
m m
j
j
Consider an arbitrary polynomial p(λ)= c j λ .By definition, p(A)= c j A . Then,
j=1 j=1
using (5.6.3.3), we get
m

p(A)= p(λ i )P i .
i=1
CAYLEY-HAMILTON THEOREM. Every normal operator satisfies its own characteristic
equation, i.e., f A (A)= O.