Suppose that the n×n matrices A and B commute; that is, that AB=BA. Prove that eA+B=eAeB.
Suppose that the n×n matrices A and B commute; that is, that AB=BA. Prove that eA+B=eAeB.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Suppose that the
n×n
matrices A and B commute; that is, that
AB=BA.
Prove that
eA+B=eAeB.
Expand the right hand side
eAeB
using the fact that if A is an
n×n
matrix, then the exponential matrix
eA
is the
n×n
matrix defined by the series
eA=
▼
Bold Upper A plus StartFraction Bold Upper A squared Over 2 exclamation mark EndFraction plus StartFraction Bold Upper A cubed Over 3 exclamation mark EndFraction midline ellipsis plus StartFraction Bold Upper A Superscript n Over n exclamation mark EndFraction plus midline ellipsis .A+A22!+A33!⋯+Ann!+⋯.
Bold Upper A plus StartFraction Bold Upper A squared Over 2 EndFraction plus StartFraction Bold Upper A cubed Over 3 EndFraction midline ellipsis plus StartFraction Bold Upper A Superscript n Over n EndFraction plus midline ellipsis .A+A22+A33⋯+Ann+⋯.
Bold Upper I plus Bold Upper A plus StartFraction Bold Upper A squared Over 2 exclamation mark EndFraction plus midline ellipsis plus StartFraction Bold Upper A Superscript n Over n exclamation mark EndFraction plus midline ellipsis .I+A+A22!+⋯+Ann!+⋯.
Bold Upper I plus Bold Upper A plus StartFraction Bold Upper A squared Over 2 EndFraction plus midline ellipsis plus StartFraction Bold Upper A Superscript n Over n EndFraction plus midline ellipsis .
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