2 × 2 Matrices That Are Roots of Unity

Alexander A. Roytvarf

A chapter in Thinking in Problems, 2013, pp 85-105 from Springer

Abstract: Abstract As readers know, a polynomial equation of degree n has at most n roots (considering multiplicity) in a number field containing its coefficients. How many roots does a polynomial equation have in a matrix ring? First, how many roots of degree n of 1=E, that is, matrices X, $$X^n = {X \circ \ldots \circ X}_n = E$$ are there? And how would one enumerate them? These and related questions that can be answered given a relatively modest level of knowledge on the reader’s part are discussed in this chapter. That is, we will characterize the roots of E in a ring of 2×2 matrices with real entries and show some applications to other mathematical topics: matrix norm estimation and spectral analysis of 3-diagonal Jacobi matrices.

Keywords: Linear Operator; Problem Group; Normed Vector Space; Jordan Canonical Form; Trigonometric Identity (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-8176-8406-8_7

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DOI: 10.1007/978-0-8176-8406-8_7

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