{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# HW3 - MATH 108B HW 3 SOLUTIONS RAHUL SHAH Problem 1 Â 7.11...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 108B HW 3 SOLUTIONS RAHUL SHAH Problem 1. [ Â§ 7.11] Solution. Assume T is a normal operator on a complex vector space. Then, by the complex spectral theorem, we find that T has an orthonormal basis consisting of eigenvectors. Thus M ( T ) is a diagonal matrix (with respect to the given orthonormal basis of eigenvectors). Let exp Ä±Î¸ j be the j-th diagonal entry of M ( T ). Let Î» j = exp Ä± Î¸ j 2 (choose one of the square roots - exists since z 2- c always has a root over the complex numbers for any c , a complex number). Let M ( T ) be a diagonal matrix with the j-th diagonal entry given by Î» j and let S be the operator that has matrix M ( S ) with respect to the given orthonormal basis of eigenvectors of T . Notice that S 2 = T . Problem 2. [ Â§ 7.12] Solution. Let V = R 2 , and let T be the operator given by a 90 â—¦ rotation. Let Î± = 0 and let Î² = 1. Then Î± 2 < 4 Î² . Notice that T 2 =- I and thus T 2 + Î±T + Î²I = 0, which is clearly not invertible. Problem 3. [ Â§ 7.13] Solution. Notice that for V , either a real or a complex vector space, and T a self-adjoint operator on T , T has an orthonormal basis consisting of eigenvectors (since every self-adjoint operator is normal, this also works for complex...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

HW3 - MATH 108B HW 3 SOLUTIONS RAHUL SHAH Problem 1 Â 7.11...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online