Canonical Forms and Solvable Singular Systems of Differential Equations

1983 ◽  
Vol 4 (4) ◽  
pp. 517-521 ◽  
Author(s):  
Stephen L. Campbell ◽  
Linda R. Petzold
Keyword(s):  
Differential Equations ◽  
Singular Systems ◽  
Canonical Forms ◽  
Systems Of Differential Equations

Indefinite inner products: Nonstandard involution

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Differential Equations ◽  
Inner Product ◽  
Canonical Forms ◽  
Inner Products ◽  
Systems Of Differential Equations ◽  
Symmetry Properties ◽  
Lagrangian Subspaces

This chapter fixes a nonstandard involution φ‎. It introduces indefinite inner products defined on Hn×1 of the symmetric and skewsymmetric types associated with φ‎ and matrices having symmetry properties with respect to one of these indefinite inner products. The development in this chapter is often parallel to that of Chapter 10, but here the indefinite inner products are with respect to a nonstandard involution, rather with respect to the conjugation as in Chapter 10. This chapter develops canonical forms for (H,φ‎)-symmetric and (H,φ‎)-kewsymmetric matrices (when the inner product is of the symmetric-type), and canonical forms of (H,φ‎)-Hamiltonian and (H,φ‎)-skew-Hamiltonian matrices (when the inner product is of the skewsymmetric-type). Applications include invariant Lagrangian subspaces and systems of differential equations with symmetries.

Indefinite inner products: Conjugation

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Differential Equations ◽  
Linear Control ◽  
Linear Control Systems ◽  
Inner Product ◽  
Canonical Forms ◽  
Linear Transformations ◽  
Inner Products ◽  
Systems Of Differential Equations ◽  
Lagrangian Subspaces ◽  
Constant Coefficients

This chapter studies matrices (or linear transformations) that are selfadjoint or skewadjoint with respect to a nondegenerate hermitian or skewhermitian inner product. As an application of the canonical forms obtained in chapters 8 and 9, canonical forms for such matrices are derived in this chapter. Matrices that are skewadjoint with respect to skewhermitian inner products are known as Hamiltonian matrices; they play a key role in many applications such as linear control systems. The canonical forms reveal invariant Lagrangian subspaces; in particular, they give criteria for existence of such subspaces. Another application involves boundedness and stable boundedness of linear systems of differential equations with constant coefficients under suitable symmetry requirements.

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