EUDML

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An envelope for the spectrum of a matrix

Panayiotis Psarrakos; Michael Tsatsomeros — 2012

Open Mathematics

We introduce and study an envelope-type region ɛ(A) in the complex plane that contains the eigenvalues of a given n×n complex matrix A. ɛ(A) is the intersection of an infinite number of regions defined by cubic curves. The notion and method of construction of ɛ(A) extend the notion of the numerical range of A, F(A), which is known to be an intersection of an infinite number of half-planes; as a consequence, ɛ(A) is contained in F(A) and represents an improvement in localizing the spectrum of A.

On the Cayley transform of positivity classes of matrices.

Fallat, Shaun M.; Tsatsomeros, Michael J. — 2002

ELA. The Electronic Journal of Linear Algebra [electronic only]

An eigenvalue inequality and spectrum localization for complex matrices.

Adam, Maria; Tsatsomeros, Michael J. — 2006

ELA. The Electronic Journal of Linear Algebra [electronic only]

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Kent Griffin; Jeffrey L. Stuart; Michael J. Tsatsomeros — 2008

Czechoslovak Mathematical Journal

Let $a$ , $b$ and $c$ be fixed complex numbers. Let $M_{n} (a, b, c)$ be the $n \times n$ Toeplitz matrix all of whose entries above the diagonal are $a$ , all of whose entries below the diagonal are $b$ , and all of whose entries on the diagonal are $c$ . For $1 \leq k \leq n$ , each $k \times k$ principal minor of $M_{n} (a, b, c)$ has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of $M_{n} (a, b, c)$ . We also show that all complex polynomials in $M_{n} (a, b, c)$ are Toeplitz matrices. In particular, the inverse of $M_{n} (a, b, c)$ is a Toeplitz matrix when...