### 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) |
---|---|

Pages (from-to) | 200-211 |

Number of pages | 12 |

Journal | Fibonacci Quarterly |

Volume | 56 |

Issue number | 3 |

State | Published - Aug 1 2018 |

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*Fibonacci Quarterly*,

*56*(3), 200-211.

**Lucas sequences and traces of matrix products.** / Greene, John R.

Research output: Contribution to journal › Article

*Fibonacci Quarterly*, vol. 56, no. 3, pp. 200-211.

}

TY - JOUR

T1 - Lucas sequences and traces of matrix products

AU - Greene, John R

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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 M1 = AB2AB2, M2 = ABAB3, and M3 = A2B4, then it is never the case that Tr(M2) > Tr(M3) > Tr(M1), 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.

AB - 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 M1 = AB2AB2, M2 = ABAB3, and M3 = A2B4, then it is never the case that Tr(M2) > Tr(M3) > Tr(M1), 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.

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M3 - Article

VL - 56

SP - 200

EP - 211

JO - Fibonacci Quarterly

JF - Fibonacci Quarterly

SN - 0015-0517

IS - 3

ER -