Suppose T element LC3 is normal and T111 222 Suppose z1 z2

Suppose T element L(C^3) is normal and T(1,1,1) = (2,2,2). Suppose (z_1, z_2, z_3) element null T. Prove that z_1 + z_2 + z_3 = 0.

Solution

Given that (1,1,1) is an eigenvector of T.with eigen value 2..

Null (T) is the set of vectors with eigenvalue 0.

Now, eigenvectors of a normal transformation corresponding to distinct eigenvectors are orthogonal.(see Theorem below)

As N is normal , (1,1,1) is orthogonal to Null(T) which implies

<(1,1,1) , (z₁, z₂, z₃) .=0 which means

z₁+ z₂+ z₃ =0

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Theorem:

Eigenvectors of a normal transformation T corresponding to distinct eigenvectors are orthogonal.

Proof:

Let Tu =pu and Tv=qv with p, q distinct,.

p<u,v>= <pu,v>

          = <Tu,v>

          = <u,T*v.> (T* being the adjoint of T)

          =<u,q*v>(q* being the complex conjugate of q)

         = q<u,v.>

          This implies (p-q) <u,v.>=0.

As p and q are distinct, it follows that <u,v>.=0,i,e u and v are orthogonal