# On the multiplicity of the eigenvalue

• krete
In summary, the conversation is about the multiplicity of the max eigenvalue of a matrix with certain conditions. The matrix has a max eigenvalue of 1 and the question is whether the multiplicity is also 1. Maxima provides the eigenvalues and their corresponding multiplicities for the matrix.

#### 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

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.

## 1. What is the multiplicity of an eigenvalue?

The multiplicity of an eigenvalue refers to the number of times an eigenvalue appears as a root of the characteristic polynomial of a matrix. In other words, it is the number of linearly independent eigenvectors associated with that eigenvalue.

## 2. How is the multiplicity of an eigenvalue related to its eigenvectors?

The multiplicity of an eigenvalue is equal to the number of eigenvectors associated with that eigenvalue. This means that an eigenvalue with a multiplicity of 2 will have 2 corresponding eigenvectors.

## 3. Can an eigenvalue have a multiplicity of 0?

No, an eigenvalue cannot have a multiplicity of 0. This is because an eigenvalue must have at least one corresponding eigenvector, and the multiplicity is equal to the number of eigenvectors.

## 4. How can the multiplicity of an eigenvalue be determined?

The multiplicity of an eigenvalue can be determined by finding the degree of its corresponding characteristic polynomial, which is equal to the number of times the eigenvalue appears as a root. Alternatively, it can also be determined by finding the number of linearly independent eigenvectors associated with that eigenvalue.

## 5. What is the significance of knowing the multiplicity of an eigenvalue?

The multiplicity of an eigenvalue is important because it provides information about the behavior and properties of a matrix. It can indicate whether a matrix is diagonalizable, and also plays a role in determining the stability of a dynamical system represented by the matrix.