Matrix pencil

S. Wold, H. Martens and H. Wold, Proc. Conf. Matrix Pencils, Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 1983, p. 286. 

A novel spectral processing approach based on information theory and the matrix pencil method was proposed by Lin et al.55 that enhances sensitivity and restores the baseline and phase properties, and hence is particularly useful, for example, for delayed acquisition experiments such as SPEDA spectra. 

The basic theory behind the generalized rank annihilation method is that the rank reduction can be re-expressed and automated. A scalar y (relative concentration of the analyte in the unknown sample) is sought such that the matrix pencil... 

Lemma 2.6.5 The matrix pencil fiE — A of the linear mechanical system (2.6.2) is regular if and only if the n x Up constraint matrix G has full rank. 

We will exclude the case of redundant constraints, so we can assume rank(G) = n and consequently all matrix pencils being regular. 

Proof As we already saw in Case 2, the Jordan canonical form plays an important role. It can be obtained by an appropriate similarity transformation applied to the system matrices. Here, a pair of matrices (E, A) must be transformed simultaneously. By this similarity transformation the matrix pencil can be transformed into the so-called Kronecker canonical form pE — A, where both, E and A, are block diagonal matrices consisting of Jordan blocks. 

Choose such that pE — A) is nonsingular. Such a p can be found, as the matrix pencil is assumed to be regular. Premultiplying the DAE by fiE — yields... 

Lemma 2.6.7 If the matrix pencil pE — A is regular, the general solution of the linear homogeneous DAE can be written as... 

We rule out the case, where the eigenvalue problem has infinitely many solutions. This is the case if the matrix pencil is singular, which corresponds to the case that the corresponding DAE is not uniquely solvable. 

Remark 2.7.4 The second part (ii) of the proof makes no use of the particular structure of and and can therefore be applied to any nonsingular matrix pencil. From the last part of the proof it can be seen that the eigenvectors corresponding to the additional zero eigenvalues are not consistent with the DAE. Thus, these eigenvalues cannot be excited by consistent initial values. 

DAEs can be classified by their index. We defined in Sec. 2.6.3 the index of a linear DAE by the index of nilpotency of the corresponding matrix pencil. For the general, nonlinear case there are many different index definitions, which might lead to different classifications in special cases. Here we will introduce the so-called perturbation index which allows to categorize the equations by means of their sensitivity under perturbations, [HLR89]. 

Wold S, Martens H, Wold H. The mnltivariate calibratitm problem in chemistry solved by the PLS method. In Rnhe A, Kagstrom B, editors. Proc. conf. matrix pencils. Heidelberg, Germania Springer Verlag 1983. p. 286, Lectnre Notes in Mathematics. 

Fokkema, D.R., Sleijpen, G.L.G., Vorst, H.A.v. Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM Journal on Scientific Computing 20(1), 94-125 (1999)... 

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