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 $\displaystyle\sum_{k,l}b_{k,l}^2=\sum_{k,l}a_{k,l}^2$.