403/503 – Another extra credit problem

This optional homework is due Thursday, April 16, at the beginning of lecture. We want to verify the details of Jacobi method for diagonalizing symmetric matrices.

Recall that the Givens rotation R(i,j,\theta) has the effect of rotating a vector by \theta radians in the i,j-plane.

Suppose A=(a_{k,l})_{k,l=1}^n is an n\times n real symmetric matrix, and that i\ne j. Let B=(b_{k,l}) be the matrix G^T A G where G=R(i,j,\theta).

  1. Show that if \theta is chosen so that \displaystyle\cos\theta=\left(\frac12+\frac{\beta}{2\sqrt{1+\beta^2}}\right)^{1/2} and \displaystyle\sin\theta=\left(\frac12-\frac{\beta}{2\sqrt{1+\beta^2}}\right)^{1/2}, where \displaystyle \beta=\frac{a_{ii}-a_{jj}}{2a_{ij}}, then b_{i,j}=b_{j,i}=0.
  2. Show that \displaystyle\sum_{k=1}^nb_{k,k}^2=2a_{i,j}^2+\sum_{k=1}^na_{k,k}^2.
  3. Show that \displaystyle\sum_{k,l}b_{k,l}^2=\sum_{k,l}a_{k,l}^2.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: