Page 244 -

5.6. LINEAR OPERATORS 211

7. Let i 1 , ... , i n be an arbitrary orthonormal basis in an n-dimensional Euclidean space V.
Then the matrix of an operator A in the basis i 1 , ... , i n is symmetric if and only if the
operator A is Hermitian.
8. In an orthonormal basis i 1 , ... , i n formed by eigenvectors of a nonnegative Hermitian
operator A, the matrix of the operator A 1/m has the form

1/m
⎛ ⎞
λ 1 0 ··· 0
⎜ 0 λ 1/m ··· 0 ⎟
⎜
⎟
2
⎜ . . . . . . . . ⎟ .
⎝ . . . . ⎠
1/m
0 0 ··· λ n
5.6.3-3. Characteristic polynomial of a linear operator.
Consider the finite-dimensional case. The algebraic equation

f A (λ) ≡ det(A – λI)= 0 (5.6.3.2)

of degree n is called the characteristic equation of the operator A and f A (λ) is called the
characteristic polynomial of the operator A.
Since the value of the determinant det(A – λI) does not depend on the basis, the
k
coefficients of λ (k = 0, 1, ... , n) in the characteristic polynomial f A (λ)are invariants
(i.e., quantities whose values do not depend on the basis). In particular, the coefficient
of λ k–1 is equal to the trace of the operator A.
In the finite-dimensional case, λ is an eigenvalue of a linear operator A if and only if λ is
a root of the characteristic equation (5.6.3.2) of the operator A. Therefore, a linear operator
always has eigenvalues.
In the case of a real space, a root of the characteristic equation can be an eigenvalue of
a linear operator only if this root is real. In this connection, it would be natural to find a
class of linear operators in a real Euclidean space for which all roots of the corresponding
characteristic equations are real.

THEOREM. The matrix A of a linear operator A in a given basis i 1 , ... , i n is diagonal if
and only if all i i are eigenvectors of this operator.

5.6.3-4. Bounds for eigenvalues of linear operators.
The modulus of any eigenvalue λ of a linear operator A in an n-dimensional unitary space
satisfies the estimate:
n n

|λ| ≤ min(M 1 , M 2 ), M 1 =max |a ij |, M 2 =max |a ij |,
1≤i≤n 1≤j≤n
j=1 i=1
where A ≡ [a ij ] is the matrix of the operator A. The real and the imaginary parts of
eigenvalues satisfy the estimates:

min (Re a ii – P i ) ≤ Re λ ≤ max(Re a ii + P i ),
1≤i≤n 1≤i≤n
min (Im a ii – P i ) ≤ Im λ ≤ max(Im a ii + P i ),
1≤i≤n 1≤i≤n