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Some Combination of $A$ and $B$ is Positive Definite

This is the general case for the definite matrix pair, and now $B$ may be singular. To be able to handle infinite eigenvalues, it is standard practice [425] to introduce a homogeneous representation of an eigenvalue $\lambda$ by a nonzero pair of numbers $(\alpha,\beta)$:

\begin{displaymath}
\lambda\equiv\alpha/\beta, \quad\quad \vert\alpha\vert^2+\vert\beta\vert^2>0.
\end{displaymath}

When $\beta = 0$, such pairs represent eigenvalue $\infty$, and this occurs when $B$ is singular. Such representations are clearly not unique since $(\xi\alpha,\xi\beta)$ represents the same ratio for any $\xi\ne 0$, and consequently the same eigenvalue. So really a pair $(\alpha,\beta)$ is a representative from a class of pairs that give the same ratio. The difference of two eigenvalues is measured by the chordal metric: for $\lambda = \alpha/\beta$ and $\wtd\lambda=\wtd\alpha/\wtd\beta$,
\begin{displaymath}
\chi(\lambda,\wtd\lambda)\equiv\chi((\alpha, \beta),(\wtd\al...
...{1 + \vert\lambda\vert^2} \sqrt{1 + \vert\wtd\lambda\vert^2}}.
\end{displaymath} (99)

An equivalent definition for a Hermitian matrix pair $\{A,B\}$ being a definite pair is that the Crawford number

\begin{displaymath}
\gamma(A,B)\equiv\min_{\Vert x\Vert _2=1}\sqrt{ (x^*Ax)^2+(x^*Bx)^2 } > 0.
\end{displaymath}

It can be proved [425] that if $\{A,B\}$ is a definite pair, then

The decompositions (5.37) and (5.38) give a complete picture of the underlying eigenvalue problems. In fact, all eigenvalues are given by pairs $(\alpha_i,\beta_i)$ with corresponding eigenvectors $Xe_i$. If, in addition, in (5.37) and (5.38) $\vert\alpha_i\vert^2 + \vert\beta_i\vert^2=1$ for all $i$, then [423]

\begin{displaymath}
\Vert X\Vert _2\le\frac 1{\sqrt{\gamma(A,B)}},\quad
\Vert X^{-1}\Vert _2\le\frac {\Vert(A,B)\Vert _2}{\sqrt{\gamma(A,B)}}.
\end{displaymath} (103)



Subsections
next up previous contents index
Next: Residual Vector. Up: Stability and Accuracy Assessments Previous: Remarks on Clustered Eigenvalues.   Contents   Index
Susan Blackford 2000-11-20