Test Matrices for the Symmetric Eigenvalue Routines

Twenty-two different types of test matrices may be generated for the symmetric eigenvalue routines. Table 1 shows the types, along with the numbers used to refer to the matrix types. Except as noted, all matrices have norm $O(1)$. The expression $U D U^{-1}$ means a real diagonal matrix $D$ with entries of magnitude $O(1)$ conjugated by a unitary (or real orthogonal) matrix $U$.

Table 1: Test matrices for the symmetric eigenvalue problem

	Eigenvalue Distribution
Type	Arithmetic	Geometric	Clustered	Other
Zero				1
Identity				2
Diagonal	3	4, 6, 7	5
$U D U^{-1}$	8, 11, 12,	9, 17$\ast$	10, 18$\ast$
	16$\ast$, 19$\star$, 20$\bullet$
Symmetric w/Random entries				13, 14, 15
Tridiagonal				21$a$
Multiple Clusters				22$b$
- matrix entries are $O(\sqrt{\mbox{overflow}})$
- matrix entries are $O(\sqrt{\mbox{underflow}})$
$\ast$ - diagonal entries are positive
$\star$ - matrix entries are $O(\sqrt{\mbox{overflow}})$ and diagonal entries are positive
$\bullet$ - matrix entries are $O(\sqrt{\mbox{underflow}})$ and diagonal entries are positive
$a$ - Some of the immediately off-diagonal elements are zero - guaranteeing splitting
$b$ - Clusters are sized: 1, 2, 4, ..., $2^i$.