The polynomial numerical hulls of Jordan blocks and related matrices

✍ Scribed by Vance Faber; Anne Greenbaum; Donald E. Marshall

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 211 KB
• Volume: 374
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00615-3

✦ Synopsis

The polynomial numerical hull of degree k for a square matrix A is a set designed to give useful information about the norms of polynomial functions of the matrix; it is defined as

|p(z)| for all p of degree k or less}.

While these sets have been computed numerically for a number of matrices, the computations have not been verified analytically in most cases.

In this paper we show analytically that the 2-norm polynomial numerical hulls of degrees 1 through n -1 for an n by n Jordan block are disks about the eigenvalue with radii approaching 1 as n → ∞, and we prove a theorem characterizing these radii r k,n . In the special case where k = n -1, this theorem leads to a known result in complex approximation theory: For n even, r n-1,n is the positive root of 2r n + r -1 = 0, and for n odd, it satisfies a similar formula. For large n, this means that r n-1,n ≈ 1log(2n)/n + log(log(2n))/n. These results are used to obtain bounds on the polynomial numerical hulls of certain degrees for banded triangular Toeplitz matrices and for block diagonal matrices with triangular Toeplitz blocks.