Matrix exponential

1 if i ≥ 0 0 otherwise
=
j =0 ∞
1 j N j! 1 j N j!
k=0 ∞
1 λk−j 1k−j (k − j )! 1 k λ = eλ k!
n−1
=
j =0
k=0
j =0
1 j N j!
or explicit eJλ = eλ I + 1 1 1 N + N2 + · · · + N n−1 , 1! 2! (n − 1)! 1 1/1! 1/(n − 1)! ... 1 λ =e .. . 1/1! 1 6
∞ k=0 ∞ k=0
ak and the limit
|ak+1 | k→∞ |ak |
ak and the limit
k
L = lim sup
k→∞
|ak |
then • If L < 1 the series converges absolutely. • If L > 1 the series diverges. If the limit is equal to 1 the series can be convergent or divergent. Theorem 1 The series (1) is convergent for all square matrix A ∈ Rn×n . Moreover eA where
n F
≤ ne
A
F
(2)
A is the Frobenius matrix norm.
i,j
2
Proof Consider the series
∞
ak
k=0
where
ak =
1 (Ak )ij k!
i.e. ak is the (i, j ) component of the matrix |Al,m | ≤ A and thus
is to find the matrix exponential of a Jordan block 0 1 1 ... ... 1 0 = λ + . . .. .. . . 1 (4) . 1 1 0 λ
1
Computing matrix exponential for diagonalizable matrices
Let be A ∈ Rn×n symmetric, then the matrix has a complete set of linear independent eigenvectors v1 , v2 , . . . , vn : Avk = λk vk , k = 1, 2, . . . , n.
2
2.1
Computing matrix exponential for general square matrices
Using Jordan normal form
Let be A ∈ Rn×n then the matrix exponential can be computed starting from Jordan normal form (or Jordan canonical form): Theorem 2 (Jordan normal form) Any square matrix A ∈ Rn×n is similar to a block diagonal matrix J , i.e. T −1 AT = J where λk 1 J1 .. J . λ 2 k and Jk = J = ... ... 1 Jm λk The column of T = [t1,1 , t1,2 , . . . , tm,nm , tm,nm −1 ] are generalized eigenvectors, i.e. Atk,j = λk tk,j λk tk,j + tk,j −1 if j = 1 if j > 1 (3)
∞ ∞ F
1 k A . It is easy to verify that k!
F
,
Ak
≤ A
k F
ak =
k=0 k=0
1 (Ak )ij ≤ k!
∞
k=0
1 Ak k!
∞ F
≤
k=0
1 A k!
k F
=e
A
F
in conclusion the series (1) is convergent for each component and inequality (2) is trivially verified.
Using A = T ΛT −1 we can write
∞
e =
k=0
A
1 k A = k!
∞
k=0
1 (T ΛT −1 )k = T k!
∞
k=0
1 k Λ k!
T −1 = T eΛ T −1 ,
and hence A e =T eλ1 e λ2 ... eλn −1 T
Thus, defining the matrix T = [v1 , v2 , . . . , vn ] whose columns are the eigenvectors we have AT = [Av1 , Av2 , . . . , Avn ] = [λ1 v1 , λ2 v2 , . . . , λn vn ] = T Λ and thus A = T ΛT −1 where Λ= λ1 λ2 .. . λn 3 .
Remark 1 (convergence criterion) here we recall some classical convergence criterion: Comparison. If ∞ k=0 bk is convergent and |ak | ≤ bk for all k ≥ n0 then ∞ k=0 ak is absolutely convergent. d’Alembert’s ratio test. Consider the series L = lim then • If the limit L exists and L < 1 the series converges absolutely. • If the limit L exists and L > 1 the series diverges. If the limit does not exist of is equal to 1 the series can be convergent or divergent. Root test. Consider the series
5
The matrix N has the property: 2 N =
0 0 0
1 .. . ...
0

1 0 0
and in general N k as ones on the k -th upper diagonal and is the null matrix if k ≥ n the dimension of the matrix. Using (4) we have
∞
e =
k=0
A
1 k 1 1 1 A = I + A + A2 + A3 + · · · + Ap + · · · k! 2 6 p!
(1)
The first question is: when the series (1) is convergent? To respond to the question we recall the following facts: 1
1 Computing matrix exponential for diagonalizable matrices 2 Computing matrix exponential for general 2.1 Using Jordan normal form . . . . . . . . . 2.2 Using Cayley–Hamilton theorem . . . . . . 2.3 Using numerical integration . . . . . . . . 2.4 Using Pade approximation and squaring . 3 square matrices 4 . . . . . . . . . . . 4 . . . . . . . . . . . 7 . . . . . . . . . . . 10 . . . . . . . . . . . 12
Matrix exponential
Enrico Bertolazzi Integration lectures for the Course: Numerical Methods for Dynamical System and Control Accademic Year 2009/2010
Contents
4
Using Jordan normal form A = T J T −1 we can write 1 1 k A = (T ΛT −1 )k e = k! k! k=0 k=0 ∞ 1 k J1 k=0 k ! ∞ 1 J2 k! k =0 =T ...
A ∞ ∞
−1 T 1 Jm k!
∞
k=0
=T Thus, the problem λ 1 λ Jλ = = λI + N