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5.2. MATRICES AND DETERMINANTS 185

Example 3. Consider the real symmetric matrix
11 –6 2
( )
A = –6 10 –4 .
2 –4 6
Its eigenvalues are λ 1 = 18, λ 2 = 6, λ 3 = 3 and the respective eigenvectors are
1 2 2
( ) ( ) ( )
X 1 = 2 , X 2 = 1 , X 3 = –2 .
2 –2 1
Consider the matrix S with the columns X 1, X 2,and X 3:
1 2 2
( )
S = 2 1 –2 .
2 –2 1
T
Taking A 1 = S AS, we obtain a diagonal matrix:
2
( 1 2 2 )( 11 –6 2 )( 1 2 2 ) ( 27 0 0 )
T
A 1 = S AS = 2 1 –2 –6 10 –4 2 1 –2 = 0 54 0 .
2
2 –2 1 2 –4 6 2 –2 1 0 0 162
–1
Taking A 2 = S AS, we obtain a diagonal matrix with the eigenvalues on the main diagonal:
2
1 ( –3 –6 –6 )( 11 –6 2 )( 1 2 2 ) ( 30 0 )
–1
A 2 = S AS =– –6 –3 6 –6 10 –4 2 1 –2 = 06 0 .
27 –6 6 –3 2 –4 6 2 –2 1 0018
2
We note that A 1 = 9A 2.
2
2
5.2.3-8. Characteristic equation of a matrix.
The algebraic equation of degree n

a 11 – λ a 12 ··· a 1n

a 21 a 22 – λ ··· a 2n
f A (λ) ≡ det(A – λI) ≡ det [a ij – λδ ij ] ≡ . . . . = 0
. . . . . . . .

a n1 a n2 ···
a nn – λ
is called the characteristic equation of the matrix A of size n × n,and f A (λ) is called its
characteristic polynomial. The spectrum of the matrix A (i.e., the set of all its eigenvalues)
coincides with the set of all roots of its characteristic equation. The multiplicity of every
root λ i of the characteristic equation is equal to the multiplicity m of the eigenvalue λ i .

i
Example 4. The characteristic equation of the matrix
4 –8 1
( )
A = 5 –9 1
4 –6 –1
has the form
4 – λ
( –8 1 )
2
3
f A(λ) ≡ det 5 –9 – λ 1 =–λ – 6λ – 11λ – 6 =–(λ + 1)(λ + 2)(λ + 3).
4 –6 –1 – λ
Similar matrices have the same characteristic equation.
Let λ j be an eigenvalue of a square matrix A.Then
1) αλ j is an eigenvalue of the matrix αA for any scalar α;
p
p
2) λ isaneigenvalue of the matrix A (p=0, 1, ... , N for a nondegenerate A;otherwise,
j
p = 0, 1, ... , N), where N is a natural number;
3) a polynomial f(A) of the matrix A has the eigenvalue f(λ).