Proving the Wielandt-Hoffman inequality

[Demon117]

Homework Statement

For a symmetric matrix A, use the notation $$\lambda_{k}\left(A\right)$$ to denote the $$k^{th}$$ largest eigenvalue, thus

$$\lambda_{n}\left(A\righ)<=...<=\lambda_{2}\left(A\right)<=\lambda_{1}\left(A\right)$$

Now suppose A and A+E are nxn symmetric matrices, prove the following results:

(a) $$\sum(\lambda_{i}\left(A+E\right)-\lambda_{i}\left(A\right))^{2}<=norm(E)_{F}^{2}$$ (Frobenius norm).

(b)$$|\lambda_{k}\left(A+E\right)-\lambda_{k}\left(A\right)|<=norm(E)_{2}$$ (2-norm) for k=1,...,n.

Homework Equations

Courant-Fischer Minimax theorem
$$\lambda_{k}\left(A\right)+\lambda_{k}\left(E\right)<=\lambda_{k}\left(A+E\right)<=\lambda_{k}\left(A\right)+\lambda_{1}\left(E\right)$$

The Attempt at a Solution

I am not really sure how to use the Courant-Fischer Minimax theorem to establish that relationship in 2. If I had some pointers I am sure I could produce something.

 

[Demon117]

you all helped a bunch...