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On the multiplicity of the eigenvalue

by krete
Tags: eigenvalue, multiplicity
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krete
#1
Dec25-09, 02:54 AM
P: 15
On the multiplicity of the eigenvalue

Dear friends,
Might you tell me any hint on the multiplicity of the max eigenvalue, i.e., one, of the following matrix.

1 0 0 0 0
p21 0 p23 0 0
0 p32 0 p34 0
0 0 p43 0 p45
0 0 0 0 0

where, pij >0 and pi,i-1 + pi,i+1 = 1

It is clear that the above matrix has a max eigenvalue one. Moreover, my numeric simulation shows that the multiplicity of the max eigenvalue seems to be 1. However, I failed to prove this observation. Might you help me? Thanks a lot.

Merry Christmas
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kakaz
#2
Dec28-09, 05:21 AM
P: 17
Quote Quote by krete View Post
On the multiplicity of the eigenvalue

Dear friends,
Might you tell me any hint on the multiplicity of the max eigenvalue, i.e., one, of the following matrix.

1 0 0 0 0
p21 0 p23 0 0
0 p32 0 p34 0
0 0 p43 0 p45
0 0 0 0 0

where, pij >0 and pi,i-1 + pi,i+1 = 1

It is clear that the above matrix has a max eigenvalue one. Moreover, my numeric simulation shows that the multiplicity of the max eigenvalue seems to be 1. However, I failed to prove this observation. Might you help me? Thanks a lot.

Merry Christmas
Maxima gives me following eigenvalues for Your matrix: [itex] [[-\sqrt{p34*p43+p23*p32},\sqrt{p34*p43+p23*p32},0,1],[1,1,2,1]] [/itex] First vector are eigenvalues, second one - multiplicity.
krete
#3
Dec28-09, 06:10 AM
P: 15
Quote Quote by kakaz View Post
Maxima gives me following eigenvalues for Your matrix: [itex] [[-\sqrt{p34*p43+p23*p32},\sqrt{p34*p43+p23*p32},0,1],[1,1,2,1]] [/itex] First vector are eigenvalues, second one - multiplicity.
thank you very much for your kind help.


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