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.
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This entry was posted on Thursday, April 9th, 2015 at 9:50 am and is filed under 403/503: Linear Algebra II. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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