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Abstract

Supervisor: Dr. B. N. MOYLS Let Mm,n (F) denote the set of all mxn matrices over the algebraically closed field F of characteristic 0, and let Mn (F) denote Mn,n (F) . Let E₃ (A) denote the third elementary symmetric function of the eigenvalues of A; let Rk = {A € Mm,n (F) : rank of A = k} ; and let mA denote the minimum polynomial of A. In this paper we are concerned with those linear transformations on Mm,n (f) for which T(Rk )⊆ Rk for various k ≤ min (m,n) ; those on Mn(f) which leave E₃ invariant; and those of Mn (F) which leave the minimum polynomial invariant. The main results are as follows: If T : Mn(F) → Mn(F) and E₃(A) = E₃(T(A)) for all A ε Mn (F) where F is the field of complex numbers, then there exist nonsingular nxn matrices U and V such that either: i) T : A → UAV for all A ε Mn (F) ; or ii) T : A → UAtV for all A ε Mn (F) ; where iii) UV = eiθIn , 3θ ≡ 0 (mod 2π) . If T : Mn(F) →Mn(F) and mA = mT(A) for all A ε Mn (F) , then T has the form i) or ii) above where UV = In. Let T : Mm,n (F) → Mm,n (F) and T(Rk) ⊆ Rk. If [formulas omitted] Then there exist nonsingular mxm and nxn matrices U [formulas omitted].