On the multiplicity of the eigenvalue

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SUMMARY

The discussion centers on the multiplicity of the maximum eigenvalue of a specified matrix, which is confirmed to be one. The matrix is characterized by non-negative elements \( p_{ij} > 0 \) and the condition \( p_{i,i-1} + p_{i,i+1} = 1 \). Numeric simulations suggest that the multiplicity of the maximum eigenvalue is one, although the original poster has not been able to prove this conclusively. A response from another participant using Maxima confirms the eigenvalues and their multiplicities, providing further insight into the matrix's properties.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors
  • Familiarity with matrix theory
  • Knowledge of numeric simulation techniques
  • Experience with Maxima or similar computational tools
NEXT STEPS
  • Research the properties of eigenvalue multiplicity in matrices
  • Explore the use of Maxima for eigenvalue calculations
  • Study numeric simulation methods for validating mathematical conjectures
  • Investigate the implications of the condition \( p_{i,i-1} + p_{i,i+1} = 1 \) on matrix behavior
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Mathematicians, researchers in linear algebra, and students studying eigenvalue problems will benefit from this discussion.

krete
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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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krete said:
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.
 
kakaz said:
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.
 

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