Problem 3. Assume that A is a nx n real matrix. We define the exponential of A, as the series 1 e^ = 1+A+ A²+₁ where I is the n x n identity matrix. a) Show that for a A-(88) А 0 b where a and b are 2 real numbers, we have 1 n! AT +... 0 *-(69). 0 eb b) Show that if B € M₂ (R) is similar to a diagonal matrix D with a transition matrix denoted by P (that is B = PDP ¹) then e³ = Pepp ¹. A c) Show that if A and B are 2 matrices such that AB = BA, then eA¹B — eªcB =
Problem 3. Assume that A is a nx n real matrix. We define the exponential of A, as the series 1 e^ = 1+A+ A²+₁ where I is the n x n identity matrix. a) Show that for a A-(88) А 0 b where a and b are 2 real numbers, we have 1 n! AT +... 0 *-(69). 0 eb b) Show that if B € M₂ (R) is similar to a diagonal matrix D with a transition matrix denoted by P (that is B = PDP ¹) then e³ = Pepp ¹. A c) Show that if A and B are 2 matrices such that AB = BA, then eA¹B — eªcB =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 3.
Assume that A is a n x n real matrix. We define the exponential of A, as the series
1
e^= 1+A+A²
where I is the n x n identity matrix.
a) Show that for
A
+.
where a and b are 2 real numbers, we have
eª
(8
=
a
= (8 %)
1
n!
0 eb
0
2).
A" +...
b) Show that if B € M₂ (R) is similar to a diagonal matrix D with a transition matrix
denoted by P (that is B = PDP ¹) then
eB Pepp 1
c) Show that if A and B are 2 matrices such that AB = BA, then eA¹B = eA eB
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