Are Eigen Vectors the Same for Eigen Values of Transpose A and A?

[nhrock3]

A is a square matrix over F field

if k is the eigen value of A



prove that k is eigen value of A^t too

and has the same eigen vectors

??



eigen vectors are the solution space P(A)

is found by solving (A-kI)x=0

dim P(A)=dim n -dim (ro(a))

rho(a)=rho(a^t)

|A|=|A^t|



these are the laws i maneged to come up with to solve it



??

 

[kru_]

We know that the eigenvalues of a matrix A are all the scalars k that satisfy the characteristic equation det( A - k In ) = 0.

We also know that det( A ) = det( A^t ) for any square matrix A.

Can we use these facts to complete a proof?

 

[nhrock3]

ok i prooved that the eigen values are the same
hw to prove that the eigen vectors are the same to for every eigen value