What is the Eigenvector of a matrix

[4 -3 1]
[-1 2 -2]
[-6 6 -4]

to find the eigenvectors of a matrix you need to calculate the characteristic polynomial of that matrix. to find the characteristic polynomial p(lambda) of a matrix you need to find the nullity of the determininate of A - lambda I_n where A is the matrix you are solving for and I_n is the identity matrix with n = dimV

V and W are vector spaces and T is a transformation such that T: V -> W is linear

then the matrix representation of T with respect to bases Beta and Gamma , which are the ordered bases of V and W respectively, = A

so okay the matrix you list transforms a vector in F^3 -> F^3 so obviously the dimension of V is 3. therefore in this case I_n = I_3

1 0 0
0 1 0
0 0 1

then Det( A - lambdaI_3) where lambda represents the variable representing eigenvalues,

= Det ( X)
X =

4 - lambda -3 1
-1 2- lambda -2
-6 6 -4 - lambda

the determinate of this is the characteristic polynomial of the matrix

now just find the roots ( zeros ) of that matrix and those are the eigenvalues of the matrix.

then just plug in those eigen values to the original equation, which is Nullity(Det(A - lambdaI_3))

and solve for this equation to get out a list of vectors, and these vectors that correspond to each eigenvalue is the eigenvector for that value.