Wednesday, 13 June 2018

linear algebra - Characteristic polynomial of a matrix 7x7?




Avoiding too many steps, which is the characteristic polynomial of this matrix 7x7? And why?



(5555555555555555555555555555555555555555555555555)


Answer



As it was stated in the commentaries, the rank of this matrix is 1; so it will have 6 null eigenvalues, which means the characteristic polynomial will be in the form:



p(λ)=αλ6(λβ)=γ6λ6+γ7λ7




Using Cayley-Hamilton:



p(A)=γ6A6+γ7A7=0



Any power of this matrix will have the same format, a positive value for all elements.



B=[1111111111111111111111111111111111111111111111111]



A=5B




A2=527B



...



A6=5675B



A7=5776B



p(A)=(γ6+35γ7)B=0γ6=35γ7




So we have: α=γ7 and β=35



p(λ)=αλ6(λ35)


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