Matrix Exponential: Putzer Formula for Variation of Parameters for ...

Matrix Exponential: Putzer Formula for eAt Variation of Parameters for Linear Dynamical Systems Undetermined Coefficients for Linear Dynamical Systems

? The 2 ? 2 Matrix Exponential eAt ? Putzer Matrix Exponential Formula for 2 ? 2 Matrices ? How to Remember Putzer's 2 ? 2 Formula

? Variation of Parameters for Linear Systems ? Undetermined Coefficients for Linear Systems

The 2 ? 2 Matrix Exponential eAt

Definition. The matrix exponential eAt is the n ? n matrix (t) defined by

(1)

d

dt

=

A,

(2) (0) = I.

Alternatively,

is the augmented matrix of

solution

vectors for

the n problems

d dt

vk

=

Avk, vk(0) = column k of I, 1 k n.

Example. A 2 ? 2 matrix A has exponential matrix eAt with columns equal to the

solutions of the two problems

d dt

v1(t)

=

Av1(t),

v1(0) =

1 0

d dt

v2(t)

=

Av2(t),

v2(0) =

0 1

Briefly, the 2 ? 2 matrix (t) = eAt satisfies the two conditions

d (1) (t) = A(t),

dt

(2) (0) =

10 01

.

Putzer Matrix Exponential Formula for 2 ? 2 Matrices

eAt

=

e1tI

+

e1t 1

- -

e2t (A

2

-

1I )

eAt = e1tI + te1t(A - 1I)

eAt

=

eat

cos

bt

I

+

eat

sin

bt (A

-

aI )

b

A is 2 ? 2, 1 = 2 real.

A is 2 ? 2, 1 = 2 real.

A is 2?2, 1 = 2 = a+ib, b > 0.

How to Remember Putzer's 2 ? 2 Formula

The expressions (1)

eAt = r1(t)I + r2(t)(A - 1I),

r1(t) = e1t,

e1t - e2t r2(t) = 1 - 2

are enough to generate all three formulas. Fraction r2 is the d/d-Newton difference quotient for r1. Then r2 limits as 2 1 to the d/d-derivative te1t. Therefore, the

formula includes the case 1 = 2 by limiting. If 1 = 2 = a + ib with b > 0, then the fraction r2 is already real, because it has for z = e1t and w = 1 the form

z - z sin bt

r2(t) = w - w =

. b

Taking real parts of expression (1) gives the complex case formula.

Variation of Parameters

Theorem 1 (Variation of Parameters for Linear Systems)

Let A be a constant n ? n matrix and F(t) a continuous function near t = t0. The unique solution x(t) of the matrix initial value problem

x (t) = Ax(t) + F(t), x(t0) = x0,

is given by the variation of parameters formula

t

(2)

x(t) = eAtx0 + eAt e-rAF(r)dr.

t0

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