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]. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/35482 |
| Date | January 1969 |
| Creators | Beasley, Leroy B |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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