State Space Form and the Drazin ODE

In the last chapters we transformed linear DAEs to ODEs the linear state space form and the Drazin ODE. Both are related to the equations of motion in DAE form in the sense, that they lead to the same solution, when initialized with corresponding initial conditions. One might wonder, if these ODEs share also other properties like stability properties How are the eigenvalues of the various forms related  

For this we will compare the eigenvalues of the state space form 

Note that, though the state space form is not unique due to the freedom in selecting y, two different state space forms always have similar system matrices Assf, so that their eigenvalues are independent of the particular choice of V. 

Theorem 2.7.3 Let A = /xi. /Xny CC be the set of eigenvalues of the state space form of the linear mechanical system E x = Asx with ny = np — n, then the following statements about the generalized eigenvalues of the different formulations hold  

Proof We prove statements (1) and (4). The others can be shown by applying similar techniques. 

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