Question

(b) The Givens rotation matrix \begin{equation} G = \begin{pmatrix} 1 & 0 & 0 \ 0 & c & s \ 0 & -s & c \end{pmatrix} \end{equation} where $c = \cos \theta$ and $s = \sin \theta$ for some $\theta$, is used to transform the matrix \begin{equation} A = \begin{pmatrix} 1 & 2 & 4 \ 2 & 3 & 0 \ 4 & 0 & 4 \end{pmatrix} \end{equation} to the tridiagonal form \begin{equation} B = GAG^ = \begin{pmatrix} b_{11} & b_{12} & 0 \ b_{21} & b_{22} & b_{23} \ 0 & b_{32} & b_{33} \end{pmatrix}. \end{equation*} Determine the Givens rotation matrix $G$ and the tridiagonal matrix $B$.

          (b) The Givens rotation matrix
\begin{equation*}
G = \begin{pmatrix}
1 & 0 & 0 \
0 & c & s \
0 & -s & c
\end{pmatrix}
\end{equation*}
where $c = \cos \theta$ and $s = \sin \theta$ for some $\theta$, is used to transform the matrix
\begin{equation*}
A = \begin{pmatrix}
1 & 2 & 4 \
2 & 3 & 0 \
4 & 0 & 4
\end{pmatrix}
\end{equation*}
to the tridiagonal form
\begin{equation*}
B = GAG^* = \begin{pmatrix}
b_{11} & b_{12} & 0 \
b_{21} & b_{22} & b_{23} \
0 & b_{32} & b_{33}
\end{pmatrix}.
\end{equation*}
Determine the Givens rotation matrix $G$ and the tridiagonal matrix $B$.

(b) The Givens rotation matrix

G =
< p m a t r i x >

where c = cosθ and s = sinθ for some θ, is used to transform the matrix

A =
< p m a t r i x >

to the tridiagonal form

B = GAG^* =
< p m a t r i x >
.

Determine the Givens rotation matrix G and the tridiagonal matrix B.

Added by David W.

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