## Abstract

Given two noncommuting matrices, A and B, it is well-known that AB and BA have the same trace. This extends to cyclic permutations of products of A's and B's. Thus if A and B are fixed matrices, then products of two A's and four B's can have three possible traces. For 2 × 2 matrices A and B, we show that there are restrictions on the relative sizes of these traces. For example, if M_{1} = AB^{2}AB^{2}, M_{2} = ABAB^{3}, and M_{3} = A^{2}B^{4}, then it is never the case that Tr(M_{2}) > Tr(M_{3}) > Tr(M_{1}), but the other five orderings of the traces can occur. By utilizing the connection between Lucas sequences and powers of a 2×2 matrix, a formula is given for the number of orderings of the traces that can occur in products of two A's and n B's.

Original language | English (US) |
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Pages (from-to) | 200-211 |

Number of pages | 12 |

Journal | Fibonacci Quarterly |

Volume | 56 |

Issue number | 3 |

State | Published - Aug 2018 |