Additive Decomposition Of Matrices With Respect To Particular Rank

Matrix decomposition is always an important topic in linear algebra. Recently, someresearchers proved that some properties of matrices can be determined by rank kmatrices, where K is a fixed positive integer. Based on this, rank-k matrix is a promisingtopic. We will investigate all the rank-k matrices in Mn(K), and we mainly focus on thata matrix can be expressed as a finite sum of rank k matrices.First, we will make an improvement for Franca Theorem. Now let’s state the theoremas follow:Franca Theorem [13, Theorem3] Let Mn(K) be the ring of all n×n matrices overthe fieldK, where charK=2,3, and Z is the center of Mn(K). Fix s∈{2,..., n1}. IfG: Mn(K)â†'Mn(K) is an additive map such that G(x)x=xG(x) for each rank-k matrixx∈Mn(K), then there exist an element λ∈Z, and an additive map μ: Mn(K)â†'Z,such that f(x)=λx+μ(x) for all x∈Mn(K).We will explain Franca Theorem by additive decomposition of matrices. And duringthe process we will find that the condition charK=2,3is not necessary.Motivated by this method, we find that a matrix can be expressed as a finite sum ofrank k matrices. Concretely, we have the following results.Theorem3.1n>1is a positive integer, Mn(K) is the ring of all n×n matrices overan arbitrary fieldK. Fix1≤r, s≤n. Then there exists a natural number t(r, s)>0determined by r and s such that for k≥t(r, s) each rank-r matrix can be expressed as a sum of k rank-s matrices, however for k <t(r, s) every rank-r matrix could not beexpressed as a sum of any k rank-s matrices. Furthermore(1) if s|r then t(r, s)=rs;(2) if r <s then t(r, s)=2;(3) if r> s and s r then t(r, s)=[rs]+1where [rs] denotes the biggest positive integernot larger thanrs.Furthermore, we use this theorem to get a property about additive map, and theconcrete result is the following theorem.Theorem3.2n>1is a positive integer, and Mn(K) is the ring of all n×n matricesover a fieldK. Then a map f: Mn(K)â†'Mn(K) must be additive if f(a+b)=f(a)+f(b)for any two invertible matrices a, b∈Mn(K).