MATH 323
Linear Algebra
Lecture 24:
Matrix polynomials.
Matrix exponentials.
Matrix polynomials
Definition. Given an n×n matrix A, we let
A2 = AA, A3 = AAA, . . . , Ak = AA
. . . A}, . . .
| {z
Also, let A1 = A and A0 = I .
k times
Associativity of matrix multiplication implies that all powers
Ak are well defined and Aj Ak = Aj+k for all j, k ≥ 0. In
particular, all powers of A commute.
Definition. For any polynomial
p(x) = c0x m + c1x m−1 + · · · + cm−1 x + cm ,
let p(A) = c0 Am + c1 Am−1 + · · · + cm−1A + cm I .
Theorem If Av = λv for a column vector v and a
scalar λ, then p(A)v = p(λ)v for any polynomial p(x).
Evaluation of a matrix polynomial is yet another problem
where the diagonalization can help.
Theorem If A = diag(a1 , a2, . . . , an ), then
p(A) = diag p(a1), p(a2), . . . , p(an ) .
Now suppose that the matrix A is diagonalizable.
Then A = UBU −1 for some diagonal matrix B and
an invertible matrix U.
A2 = UBU −1UBU −1 = UB 2U −1,
A3 = A2 A = UB 2U −1UBU −1 = UB 3 U −1.
Likewise, An = UB n U −1 for any n ≥ 1.
I + 2A − 3A2 = UIU −1 + 2UBU −1 − 3UB 2U −1
= U(I + 2B − 3B 2)U −1.
Likewise, p(A) = Up(B)U −1 for any polynomial p(x).
• Initial value problem for a linear ODE:
dy
dt
= 2y ,
y (0) = 3.
Solution: y (t) = 3e 2t .
• Initial value problem for a system of linear ODEs:
(
dx
= 2x + 3y ,
dt
x(0) = 2, y (0) = 1.
dy
= x + 4y ,
dt
The system can be rewritten in vector form
d x
x
2 3
=A
, where A =
.
y
1 4
dt y
x(t)
2
tA
Solution:
=e
.
y (t)
1
What is e tA ?
Exponential function
Exponential function: f (x) = exp x = e x , x ∈ R.
Principal property: e x+y = e x · e y .
x n
x
.
Definition 1. e = lim 1 +
n→∞
n
1 n
In particular, e = lim 1 +
≈ 2.7182818.
n→∞
n
xn
x2
+ ··· +
+ ···
Definition 2. e = 1 + x +
2!
n!
x
Definition 3. f (x) = e x is the unique solution of
the initial value problem f 0 = f , f (0) = 1.
Complex exponentials
Definition. For any z ∈ C let
zn
z2
z
+ ··· +
+ ···
e =1+z +
2!
n!
Remark. A sequence of complex numbers
z1 = x1 + iy1, z2 = x2 + iy2, . . . converges
to z = x + iy if xn → x and yn → y as n → ∞.
Theorem 1 If z = x + iy , x, y ∈ R, then
e z = e x (cos y + i sin y ).
In particular, e iφ = cos φ + i sin φ, φ ∈ R.
Theorem 2 e z+w = e z · e w for all z, w ∈ C.
Proposition e iφ = cos φ + i sin φ for all φ ∈ R.
Proof: e
iφ
(i φ)2
(i φ)n
= 1 + iφ +
+ ··· +
+ ···
2!
n!
The sequence 1, i , i 2, i 3, . . . , i n , . . . is periodic:
1, i , −1, −i , 1, i , −1, −i , . . .
| {z } | {z }
It follows that
φ2 φ4
φ2k
e iφ = 1 −
+
− · · · + (−1)k
+ ···
2!
4!
(2k)!
2k+1
φ
φ3 φ5
+
− · · · + (−1)k
+ ···
+ i φ−
3!
5!
(2k + 1)!
= cos φ + i sin φ.
Matrix exponentials
Definition. For any square matrix A let
1
1
exp A = e A = I + A + A2 + · · · + An + · · ·
2!
n!
Matrix exponential is a limit of matrix polynomials.
Remark. Let A(1) , A(2) , . . . be a sequence of n×n
(n)
matrices, A(n) = (aij ). The sequence converges to
(n)
an n×n matrix B = (bij ) if aij → bij as n → ∞,
i.e., if each entry converges.
Theorem The matrix exp A is well defined, i.e.,
the series converges.
Properties of matrix exponentials
Theorem 1 If AB = BA then e A e B = e B e A = e A+B .
Corollary (a) e tA e sA = e sA e tA = e (t+s)A , t, s ∈ R;
(b) e O = I ; (c) (e A )−1 = e −A .
Theorem 2
d tA
e = Ae tA = e tA A.
dt
tn n
t2 2
Indeed, e = I + tA + A + · · · + A + · · · ,
2!
n!
and the series can be differentiated term by term.
d tn n
d tn
t n−1
n
A
An .
=
A =
dt n!
dt n!
(n − 1)!
tA
Lemma Let A be an n×n matrix and x ∈ Rn .
Then the vector function v(t) = e tA x satisfies
v0 = Av.
dv
d
Proof:
= e tA x = Ae tA x = A e tA x = Av.
dt
dt
Theorem For any t0 ∈ R and x0 ∈ Rn the initial
value problem
dv
= Av, v(t0) = x0
dt
has a unique solution v(t) = e (t−t0 )A x0 .
Indeed, v(t) = e (t−t0 )A x0 = e tA e −t0 A x0 = e tA x,
where x = e −t0 A x0 is a constant vector.
Evaluation of matrix exponentials
Example. A = diag(a1 , a2, . . . , ak ).
An = diag(a1n , a2n , . . . , akn ), n = 1, 2, 3, . . .
e A = I + A + 2!1 A2 + · · · + n!1 An + · · ·
= diag(b1 , b2, . . . , bk ),
where bi = 1 + ai + 2!1 ai2 + 3!1 ai3 + · · · = e ai .
Theorem 1 If A = diag(a1 , a2, . . . , ak ) then
e A = diag(e a1 , e a2 , . . . , e ak ),
e tA = diag(e a1 t , e a2 t , . . . , e ak t ).
Theorem 2 If A = UBU −1, then e A = Ue B U −1.
Moreover, tA = U(tB)U −1 so that e tA = Ue tB U −1.
Example. A =
2 3
.
1 4
The eigenvalues of A: λ1 = 1, λ2 = 5.
3
1
Eigenvectors: v1 =
, v2 =
.
−1
1
Therefore A = UBU −1
, where 3 1
1 0
, U=
.
B=
0 5
−1 1
Then
t
−1
3
1
e
0
3
1
e tA = Ue tB U −1 =
−1 1
0 e 5t
−1 1
t 5t t
3e e
3e + e 5t −3e t + 3e 5t
1 1 −1
1
=
=4
.
−e t e 5t 4 1 3
−e t + e 5t
e t + 3e 5t
Problem. Solve a system of differential equations
(
dx
dt = 2x + 3y ,
dy
dt
= x + 4y
subject to initial conditions x(0) = 2, y (0) = 1.
The unique solution:
x(t)
2
= e tA x0, where A =
y (t)
1
t
3e + e 5t −3e t
e tA = 14
−e t + e 5t
et
(
x(t) = 34 e t + 54 e 5t ,
=⇒
y (t) = − 41 e t + 54 e 5t .
3
2
, x0 =
.
4
1
+ 3e 5t
+ 3e 5t


0 1 0
Example. A = 0 0 1, a Jordan block.
0 0 0


0 0 1
2
A = 0 0 0, A3 = O, An = O for n ≥ 3.
0 0 0


1 1 21
1
e A = I + A + A2 = 0 1 1,
2
0 0 1


1 t 12 t 2
2
t
e tA = I + tA + A2 = 0 1 t .
2
0 0 1
λ 1
Example. A =
, a general Jordan block.
0 λ
0 1
We have that A = λI + B, where B =
.
0 0
Since (λI )B = B(λI ), it follows that e A = e λI e B .
Similarly, e tA = e tλI e tB .
λt
e
0
e tλI =
= e λt I ,
0 e λt
1
t
B 2 = O =⇒ e tB = I + tB =
.
0 1
λt
λt
e
te
Thus e tA = e tλI e tB =
.
0 e λt
Problem. Solve a system of differential equations
(
dx
dt = 2x + y ,
dy
dt
= 2y
subject to initial conditions x(0) = y (0) = 1.
The unique solution:
x(t)
2 1
1
tA
= e x0, where A =
, x0 =
.
y (t)
0 2
1
2t
2t
1
t
e
te
= e 2t
e tA =
0 e 2t
0 1
(
x(t) = e 2t (1 + t),
=⇒
y (t) = e 2t .