may or may not exist. The number Nof linearly independent eigenvectors associated
with an eigenvalue λiis given by the formula
N=n−(rank [A−λiI]),
where nis the rank of A.
Infinite Series of Square Matrices
In the following discussions it is to be understood that the matrix Ais an n×n
constant square matrix with eigenvalues λ 1 ,... , λ nwhich are distinct. Consider the
infinite series
S(x) = c 0 +c 1 x+c 2 x^2 +···+ckxk+··· =
∑∞
k=0
ckxk (10 .35)
and the corresponding matrix infinite series
S(A) = c 0 I+c 1 A+c 2 A^2 +···+ckAk+··· =
∑∞
k=0
ckAk. (10 .36)
where the n×n matrix A has replaced the value x in equation (10.35) and the
identity matrix has replaced the coefficient of the c 0 constant term. Convergence of
the matrix infinite series can be defined in a manner analogous to that of the scalar
infinite series. Examine the sequence of partial sums
SN=
∑N
k=0
ckAk
and if lim N→∞ SN exists, then the matrix series is said to converge, otherwise it is
said to diverge. It can be shown that if the series in equation (10.35) is convergent
for x=λi (i= 1 , 2 ,... , n ),where λi is an eigenvalue of A, then the matrix series in
equation (10.36) is convergent.
Some specific examples of series associated with a n×nconstant square matrix
Aare the following.
1. The Exponential Series Corresponding to the scalar exponential series
ex t = 1 + x t +x^2 t
2
2!
+···+xkt
k
k!
+···
there is the exponential matrix eA t defined by the series
eA t =I+A t +A^2
t^2
2! +···+A
ktk
k!+··· (10 .37)
