The eigenvalues of Aare determined by the characteristic equation
C(λ) = det (A−λI ) = |A−λI |= (−1)nλn+α 1 λn−^1 +··· +αn− 1 λ+αn= 0
If λ= 0 is a root of the characteristic equation, then C(λ) = det A= 0 and
consequently the matrix Ais singular.
Property 4: Two matrices Aand Bare said to be similar if there exists
a nonsingular matrix Qsuch that B=Q−^1 AQ. If the matrices Aand B
are similar, then they have the same characteristic equation.
The above property is established if it can be shown that the character-
istic equation of Bequals the characteristic equation of A. For Qnonsingular
and B=Q−^1 AQ , one can write
B−λI =Q−^1 AQ −λI
=Q−^1 AQ −λQ −^1 IQ
=Q−^1 (A−λI )Q.
The determinant of this equation gives
C(λ) = |B−λI |=
∣∣
Q−^1 (A−λI )Q
∣∣
=
∣∣
Q−^1
∣∣
|A−λI ||Q|.
But QQ −^1 =Iand |Q|
∣∣
Q−^1
∣∣
= 1 hence C(λ) = |B−λI |=|A−λI |and the above property
is established.
Additional Properties Involving Eigenvalues and Eigenvectors
The following are some additional properties and definitions relating to
eigenvalues and eigenvectors of an n×nsquare matrix A. The properties are
given without proof.
1. If the neigenvalues λ 1 , λ 2 ,... , λ nof Aare all distinct, then there exists
n-linearly independent eigenvectors.
2. If an eigenvalue repeats itself, then the characteristic equation is said to
have a multiple root. In such cases there may or may not exit nlinearly
independent eigenvectors. If the characteristic equation can be written
C(λ) = (λ−λ 1 )n^1 (λ−λ 2 )n^2 ···(λ−λk)nk= 0
where
∑k
i=1 ni=n, then niis called the multiplicity of the eigenvalue λi.
