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\title{3-D Jacobi Rotation Method for Eigenvalue Problems of 
Matrices}

\author{Da Yong Cai \\
Tsinghua University, Beijing, China \\
\and
Weimin Xue\\
Hong Kong Baptist University and Tsinghua University}

\date{}
\maketitle


The eigenproblems for matrices are important for both scientific research
and engineering purposes. Up to now, the most effective algorithm is the
QR method with shift strategy. The computations show that the QR method
is less favorable if the matrix has clustered spectrum. Therefore it is
still worth-while to explore some new approaches for the eigenvalues of
matrices with clustered spectrum.

\vskip 2truemm

On the other hand, from Schur decomposition theorem, we know that there
exists an orthogonal matrix $Q\subset {\bf R}^{n\times n}$ such that
$
Q^T A \hspace*{1mm} Q = T
$
is a block triangular matrix, in which  each
main diagonal block of $T$ is either ${\bf R}^{1\times 1}$ or
${\bf R}^{2\times 2}$.
To our best knowledge, it is still a task to find the numerical algorithm
to implement the Schur decomposition effectively.

\vskip 2truemm 

The Jacobi method is one of the oldest numerical method for the eigenvalue
problems of symmetric matrices. 
It radiated with a brief light in the 1950s, and, in recent years,
has comeback because its adaptability to parallel computers.

\vskip 2truemm 

The basic idea of the Jacobi method is to find a $2\times2$ Jacobi rotator 
$Q$ for the $2\times2$ symmetric matrix $A$ such that 
$Q^T A \hspace*{1mm} Q$ is diagonal.
The Jacobi rotation method is essentially not suitable for 
non-symmetric matrices.

\vskip 2truemm 

The basic idea of the 3-D Jacobi method is to find two $3\times3$ Jacobi 
rotators $Q_1$ and $Q_2$ for the $3\times3$ matrix $A$ such that 
the matrix A becomes a Hessenberg matrix after two rotations, i.e.
$$
Q_2^T Q_1^T A \hspace*{1mm} Q_1 Q_2=
\left[
\begin{array}{lll}
{\tilde a}_{11} & {\tilde a}_{12} & {\tilde a}_{13} \\
              0 & {\tilde a}_{22} & {\tilde a}_{23} \\
              0 & {\tilde a}_{32} & {\tilde a}_{33} . \\
\end{array}
\right]
$$

In this paper we will dicuss the  formulation of the 3-D Jacobi 
rotation method in an algebraic deduction and report some numerical
results for symmetric
matrices with clusted spectrum 
and Schur decomposition for non-symmetric matrices.

\bigskip

%
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