The envelope of tridiagonal Toeplitz matrices and block-shift matrices

The envelope of a square complex matrix is a spectrum encompassing region in the complex plane. It is contained in and is akin to the numerical range in the sense that the envelope is obtained as an infinite intersection of unbounded regions contiguous to cubic curves, rather than half-planes. In this article, the geometry and properties of the envelopes of special matrices are examined. In particular, symmetries of the envelope of a tridiagonal Toeplitz matrix are obtained, and the envelopes of block-shift matrices, Jordan blocks and 2×2 matrices are explicitly characterized.