Odd order absolute cosine powers

Abstract

Absolute cosine matrix powers and their Fourier modes. The matrix and conjecture are those of Jain.

Definition 1.1 (Angular step).

Source. Repository-derived.

Acknowledgement. Tanvi Jain (2017). Hadamard powers of some positive matrices. DOI: 10.1016/j.laa.2016.06.030. URL: https://doi.org/10.1016/j.laa.2016.06.030.

Commentary.

For order 2m+1 the angular step is pi divided by 2m+1.

Definition 1.2 (Entrywise real powers).

Source. Repository-derived.

Acknowledgement. Tanvi Jain (2017). Hadamard powers of some positive matrices. DOI: 10.1016/j.laa.2016.06.030. URL: https://doi.org/10.1016/j.laa.2016.06.030.

Commentary.

The matrix entry at a,b is the real power r of the absolute value of cos((a-b) pi/(2m+1)).

Definition 1.3 (Real Fourier mode).

Source. Repository-derived.

Acknowledgement. Tanvi Jain (2017). Hadamard powers of some positive matrices. DOI: 10.1016/j.laa.2016.06.030. URL: https://doi.org/10.1016/j.laa.2016.06.030.

Commentary.

The j-th mode is one plus twice the sum over l from one to m of cos(2jl pi/(2m+1)) times cos(l pi/(2m+1)) to the real power r.

Definition 1.4 (Even root product).

Source. Repository-derived.

Acknowledgement. Tanvi Jain (2017). Hadamard powers of some positive matrices. DOI: 10.1016/j.laa.2016.06.030. URL: https://doi.org/10.1016/j.laa.2016.06.030.

Commentary.

The product over s less than j of r-2s records the proposed roots and their signs.

Definition 1.5 (Nonzero eigenvalue multiplicity).

Source. Repository-derived.

Acknowledgement. Tanvi Jain (2017). Hadamard powers of some positive matrices. DOI: 10.1016/j.laa.2016.06.030. URL: https://doi.org/10.1016/j.laa.2016.06.030.

Commentary.

For m at least one and nonnegative r, every nonzero root of the characteristic polynomial other than mode zero has root multiplicity exactly two.