scholarly journals Simultaneous diagonalization of incomplete matrices and applications

2020 ◽  
Vol 4 (1) ◽  
pp. 127-142
Author(s):  
Jean-Sébastien Coron ◽  
Luca Notarnicola ◽  
Gabor Wiese
Keyword(s):  
Simultaneous Diagonalization

scholarly journals A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application

1970 ◽  
Vol 11 (1) ◽  
pp. 81-83 ◽  
Author(s):  
Yik-Hoi Au-Yeung
Keyword(s):  
Complex Numbers ◽  
Single Element ◽  
Singular Matrix ◽  
Necessary And Sufficient Condition ◽  
Simultaneous Diagonalization ◽  
Hermitian Matrices ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Necessary And Sufficient ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

On the Connection of Nonrenormalizable and Renormalizable Unified Nonlinear Spinor Field IVfodels

1984 ◽  
Vol 39 (5) ◽  
pp. 441-446
Author(s):  
H. Stumpf
Keyword(s):  
Field Equation ◽  
Spinor Field ◽  
Order Derivative ◽  
Simultaneous Diagonalization ◽  
Field Equations ◽  
Nonlinear Spinor ◽  
Original Field ◽  
Higher Order Derivative ◽  
Grand Canonical ◽  
Spinor Field Equation

The nonrenormalizable first order derivative nonlinear spinor field equation with scalar interaction possesses two equivalent Hamiltonians. The first is the conventional one while the second is a two-field Hamiltonian with the original field and its parity transform. By quantization the latter leads to an inequivalent representation compared with the former. This is connected with parity symmetry breaking and the loss of simultaneous diagonalization of energy and subfield particle numbers. The corresponding grand canonical Hamiltonian is shown to result equivalently from a renormalizable second order derivative nonlinear spinor field equation. This is achieved by means of a theorem about the decomposition of higher order derivative nonlinear spinor field equations derived previously

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