Algebraic Multiplicity of an Eigenvalue

[vish_maths]

Please have a look at the attached images.I am attempting a proof for the statement : The algebraic multiplicity of an eigen value λ is equal to dim null [T - λ I] ^{dim V}.

Please advise me on how to move ahead. Apparently, I am at the final inference required for a proof but unable to move ahead. Thank you very much for your help in this regard.

Reference : To know how results (2) and (3) in the images come, you may please have a look at this ebook - pg 165, http://fetweb.ju.edu.jo/staff/EE/jrahhal/PDF/sc%20-%20Linear%20Algebra%20Done%20Right.pdf

 

[WWGD]

In my experience, the definition you gave is of the _geometric _ multiplicity, not the algebraic one. The two multiplicities may or may not coincide.

 

[vish_maths]

In my experience, the definition you gave is of the _geometric _ multiplicity, not the algebraic one. The two multiplicities may or may not coincide.

Thank you for the answer. The notes which I have say that the number of times λ appears on the diagonal of an upper triangular matrix is equal to dim null [T - λ I] ^{dim V}. Do you think there is error in this statement as well?"

I have actually attempted the proof to prove this statement. Sorry about incorrectly saying it equal to algebraic multiplicity

 

[WWGD]

Ah, sorry, I did not see the exponent dimV in your post , at first. In my experience, the geometric multiplicity of $\lambda$ is the dimension of null$[T- \lambda I]$. The algebraic multiplicity of $\lambda$ is the multiplicity of the exponent of $\lambda$ in the factorization of $A-cI$.

 

[vish_maths]

No problem :) . Do you think I made a good attempt at the proof? I just think i got stuck in the last stage of the proof