On the multiplicity of the eigenvalue

[krete]

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

 

[kakaz]

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: $[[-\sqrt{p34*p43+p23*p32},\sqrt{p34*p43+p23*p32},0,1],[1,1,2,1]]$ First vector are eigenvalues, second one - multiplicity.

 

[krete]

Maxima gives me following eigenvalues for Your matrix: $[[-\sqrt{p34*p43+p23*p32},\sqrt{p34*p43+p23*p32},0,1],[1,1,2,1]]$ First vector are eigenvalues, second one - multiplicity.

thank you very much for your kind help.