Fibonacci vector and matrix <i>p</i> -norms

Abstract This article delves into vector and matrix norms of Fibonacci numbers. Two classes of Fibonacci vectors and a parametric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -norm are defined. From this definition, several properties of the Fibonacci vector and matrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -norms are described by varying parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p . A closed-form expression is given to obtain the value of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p , setting the difference between the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -norm and the infinite norm below a given threshold. A new symmetric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -Fibonacci matrix is defined, and a simple reorganization simplifies the computation of its <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -norm. The analysis is extended to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -distances when considering the norm of the difference of two vectors (matrices) of the same size.