For the linear operator T on an inner product space V, determine whether T is no

Question

For the linear operator T on an inner product space V, determine whether T is normal, self-adjoint or neither. If possible, produce an orthonormal basis of eigenvectors of T for V and list the eigenvalues. PLEASE EXPLAIN STEP BY STEP !!!!!!!

1. V=C^2 and T is defined by T ( a , b ) = ( 2a + ib, a + 2b )

So I know that (AxA*)=(A*xA) and that T is normal but not self-adjoint. I cannot figure out how to compute the orthonormal basis because when I do ( A - t I) to find the eigenvalues and eigenvectors I get the following:    

( 2 - t )^2 - i = t^2 - 4t + 4 - i

I know the answer is: (orthonormal basis is) : ( 1/(sqrt2) , -1/2 + 1/2 i ) , ( 1/(sqrt2) , 1/2 - 1/2 i)

HOW DO I DO THIS? STEP BY STEP PLEASE

THANKS

Explanation / Answer

The transformation T can be described by the matrix [2 i // 1 2]. So find the orthonormal eigenvectors for that (and the corresponding eigenvalues) and you have the orthonormal basis for part a. Using good ol' MATLAB ("eig" function), you get

[0.5+0.5i // 0.5 * sqrt(2)],

and [ 0.5*sqrt(2) // -0.5+0.5i]

these two eigen vectors are orthogonal and normalized, so they are an orthonormal basis that spans the resulting space after V is mapped through T.

The corresponding eigenvalues (from MATLAB) are
2+0.5*sqrt(2)+0.5*sqrt(2)*i, and
2-0.5*sqrt(2)-0.5*sqrt(2)*i.

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects