Matrix Spaces Preserving Rank Export. Domain Mapping Applications

The matrix algebra as an immportant research area of algebra has wide applica-tions in the areas such as geometry, graph theory, economics, engineering, statistics,etc. Linear preserver problems(LPPs for short) has been a very active branch ofmatrix algebra, since many excellent results have appeared during the last 30 years.Let F be a field, f_ij (i, j∈[n]) the maps on F, S_n(F) the space of all symmetricmatices on F. Let f: S_n(F)→S_n(F) be a map induced by {f_ij}_n. In this thesis,the induced maps preserving rank-1 on symmetric matices are characterized. Weprove the following theorem.Theorem Let F be a field, n is integer with n＞2. and f: S_n(F)→S_n(F)induced by {f_ij}_n be an rank one preserver. Then f is of one of the following forms:(â…°) There exists a r₀∈[n] such that f(X)=f_r0r0(X_r0r0)E_r0r0, (?)X=(x_ij)_n×n∈S_n(F),and f_r0r0 satisfies that f_r0r0(x)≠0, (?)x∈F;(â…±) There exist s₀, t₀∈[n](s₀≠t₀) and c₁, c₂∈F^*, such that for anyX=(x_ij)_n×n∈S_n(F), f(X)=c₁E_s0s0+f_s0t0(x_s0t0)E_s0t0+f_s0t0(x_s0t0)E_t0s0+c₂E_t0t0,and f_s0t0 satisfies: f_s0t0²(x)=c₁c₂, (?)x∈F;(â…²) There exists M∈S_n(F), rankM=1 such that f(X)=M, (?)X∈S_n(F).(â…³) There existα∈F^* and P∈GL_n(F) where P is a diagonal matrice, suchthat f(X)=αPX^φP^T, (?)X∈S_n(F). whereφ: F→F is an multiplicative map which satisfied thatφ(x)=0(?)x=0.Thereby, we give a necessary and sufficient conditions under which an inducedmap is a rank preserver, i.e.,Theorem Let F be a field, n is integer with n＞2. and f: S_n(F)→S_n(F)be a map induced by {f_ij}_n. Then f is an rank preserver if and only if there existα∈F^* and P∈GL_n(F) where P is a diagonal matrice, such that f(X)=αPX^φP^T, (?)X∈S_n(F).whereφ: F→F is an nonzero injective homorphism.