Math Help - eigenvector u+v

1. eigenvector u+v

let A be an nxn matrix and suppose that u and v are eigenvectors of A,both with corresponding eigenvalue y.Prove that if u+v is not equal to 0n , then u+v is also an eigenvector of A with corresponding eigenvalue y

2. Originally Posted by math8553
let A be an nxn matrix and suppose that u and v are eigenvectors of A,both with corresponding eigenvalue y.Prove that if u+v is not equal to 0n , then u+v is also an eigenvector of A with corresponding eigenvalue y
By definition $A\bold{u} = y\bold{u}$ and $A\bold{v} = y\bold{v}$.
This means, $A\bold{u} + A\bold{v} = y\bold{u}+y\bold{v}\implies A(\bold{u}+\bold{v}) = y(\bold{u}+\bold{v})$
Thus, if $\bold{u}+\bold{v}\not = 0$ then it is an eigenvector with eigenvalue $y$.