P. Van Dooren, The computation of Kronecker's canonical form of a singular pencil, Linear Algebra Appl., 27 (1979), pp. 103--140.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the staircase algorithm for computing the Jordan structure of a multiple eigenvalue by unitary similarity transformations. This milestone contribution stimulated numerous subsequent papers on algorithms for the numerical computation of Jordan and Kronecker structure information (e.g. see [40, 29, 28, 41, 26, 3, 11, 12, 30]) including several important contributions by Kublanovskaya herself (e.g. see [34, 35, 31, 32, 4, 36, 37] Moreover, new insight and understanding of the mathematical theory of orbits and bundles of matrices and matrix pencils have led to new results, which, in turn, has stimulated the ....

....n) The KCF looks quite complicated in the general case, but most matrix pencils have a more simple Kronecker structure. Almost all rectangular m n pencils A #B (m n) have the same KCF, depending only on m and n, and this KCF only includes the blocks L j if m nand L j otherwise (e.g. see [41, 10, 14]) This corresponds to the generic case where A #B has full rank for any scalar #. It follows that generic rectangular pencils have no regular part. We note that any pencil with only L j or only L j blocks has full rank, but it is only one of them which is the generic structure. Square pencils ....

P. Van Dooren, "The computation of Kronecker's canonical form of a singular pencil," Linear Algebra Appl., 27, 103--141 (1979).

.... k columns of Q, respectively. When 1 is empty, the corresponding reducing subspaces are called minimal, and when 1 contains the whole spectrum the reducing subspaces are called maximal. Several authors have proposed (staircase type) algorithms for computing a generalized Schur form (e.g. see [2, 22, 24, 23, 31, 36]) They are numerically stable in the sense that they compute the exact Kronecker structure (generalized Schur form or something similar) of a nearby pencil A Gamma B . ffi j k(A Gamma A ; B Gamma B )k E is an upper bound on the distance to the closest (A ffiA; B ffiB) with the ....

....rectangular pencils have no regular part. The generic Kronecker structure for A Gamma B with d = n Gamma m 0 is diag(L ff ; L ff ; L ff 1 ; L ff 1 ) where ff = bm=dc, the total number of blocks is d, and the number of L ff 1 blocks is m mod d (which is 0 when d divides m) [31, 8]. The same statement holds for d = m Gamma n 0 if we replace L ff ; L ff 1 in (3.2) by L ff ; L ff 1 . Square pencils are generically regular, i.e. det(A Gamma B) 0 if and only if is an eigenvalue. The generic singular pencils of size n by n have the Kronecker structures [34] diag(L j ....

P. Van Dooren. The computation of Kronecker's canonical form of a singular pencil. Lin. Alg. Appl., 27:103--141, 1979.

....the finite or infinite structure of a pencil from its Smith form is not numerically reliable. Instead of that, we will obtain this information in a numerically reliable way from the Kronecker invariants of suitable matrix pencils. The algorithm for that purpose is based on the results from [17, 18] and uses only numerically reliable operations such as Householder transformations or the singular value decomposition (SVD) see for instance [7, 8] Let us consider an arbitrary m n matrix A and compute its SVD, AQ = # where # is an m n matrix with singular values of A along its main ....

....(9b) where A r and A c have r linearly independent rows and columns respectively. Operation (9a) is called row compression and operation (9b) column compression of A. Consider an arbitrary m n pencil = sE L. The algorithm to obtain the eigenstructure of is described as follows (see [17]) Let E 1 = E, n 1 = n, m 1 = m and L 1 = L. Step k: Obtain the SVD # = P E k Q and the rank # k of L k . If v k = m k # k is not zero, make the permuted row compressions. E k L k L k where I p = 0 I #k I vk 0 . Obtain the SVD S = P L k Q and the rank r k ....

Van Dooren P., "The computation of Kronecker's canonical form of a singular pencil", Linear Algebra an its Applications, 27, 103-141 (1979).

....computed eigenvalues correspond to perturbed multiple eigenvalues (ii) computing the structure : given a restored multiple eigenvalue, find out what its Jordan or Kronecker structure is. 1 Both the algorithmic aspects of these two problems (i.e. issues as complexity and stability [14] 15] 9] [17] [10] 11] 2] and their theoretical aspects (i.e. issues as sensitivity and robustness [13] 5] 20] 21] 6] 7] have extensively been treated in the literature during the last 20 years. In this paper we focus on the second problem and more specifically on the reconstruction of the Jordan ....

....extensively been treated in the literature during the last 20 years. In this paper we focus on the second problem and more specifically on the reconstruction of the Jordan structure of a matrix A at a given eigenvalue. This problem is known to be equivalent to a recursive rank search [14] 15] [17], during which the Jordan structure of the corresponding eigenvalue is being reconstructed throughout the recursion. It turns out that in the worst case one has to perform (n Gamma 1) rank decompositions of matrices of decreasing dimension (n Gamma i) i = 1; n Gamma 1, hence leading to ....

P. Van Dooren, The computation of Kronecker's canonical form of a singular pencil, Linear Algebra & Applications 27 (1979) 103-140.

....as Kronecker invariants of the system. However, from the linear system point of view, it is interesting to obtain this complete set of invariants in terms of the system itself. This has been accomplished by Morse [9] under the non essential restriction D = 0, and Molinari [8] We refer also to [10], 7] and [6] for the computation of these invariants and [4] where this problem it treated from the point of view of the (P; Q) blocks. Here we present an geometrical approach extending that of [2] where is treated the case of pairs (A; B) This approach allow to compute beside a complete set ....

P. Van Dooren. The Computation of Kronecker's Canonical Form of a Singular Pencil. Lin. ALg. Appl. vol.27, pp.103-140, 1970. 28

.... proposed in [8] In the numerical linear algebra literature, several algorithms based on generalized state space realizations and the correspondance between the Smith MacMillan form of a rational matrix and the Kronecker canonical form of an associated pencil matrix were proposed by Van Dooren [18]. These pencil algorithms allow to compute at once the whole structure (finite and infinite) of a polynomial matrix [19] In contrast, the algorithm proposed in our paper is not based on the pencil matrix associated with the input polynomial matrix. It is rather similar in spirit to the algorithm ....

....(orthogonal) transformations. Moreover, the routines takeadvantage of the special structure of the matrices so as to reduce the overall computational cost. In this regard, our algorithm can thus be viewed as a polynomial approach alternative to the state space approach algorithms proposed in [18]and recently implemented in the Fortran library Slicot, see [21] The numerical routines described in this paper are implemented in the new release 3.0 of the Polynomial Toolbox for Matlab [15] 2 Infinite Zeros and Infinite Structure In this section, we recall standard facts on the ....

P.Van Dooren "The Computation of Kronecker's Canonical Form of a Singular Pencil", Linear Algebra and Its Applications, Vol. 27, pp. 103--141, 1979.

....looks quite complicated in the general case, but most matrix pencils have a more simple Kronecker structure. Almost all rectangular m Theta n pencils A Gamma B (m 6= n) have the same KCF, depending only on m and n and this KCF only includes L j (if m n) and L T j blocks otherwise (e.g. see [26, 6, 11]) This corresponds to the generic case when A Gamma B has full rank for any scalar . It follows that generic rectangular pencils have no regular part. We remark that any pencil with only L j or only L T j blocks has full rank, but it is only one of them which is the generic structure. Square ....

P. Van Dooren. The computation of Kronecker's canonical form of a singular pencil. Lin. Alg. Appl., 27:103--141, 1979.

....matrices X and Y such that X T EY = E 1 E 2 ; X T AY = A 1 A 2 ; 2.1) where E 1 is nonsingular, A 2 is lower triangular and nonsingular, E 2 is strictly lower triangular, i.e. lower triangular with all diagonal entries being zero. This can be done by Van Dooren s algorithm [7] to which we shall return later. Then H(s) D C T Y (sX T EY Gamma X T AY ) Gamma1 X T B: Partition conformally X T B = B 1 B 2 ; Y T C = C 1 C 2 : We have H(s) D C T 1 (sE 1 Gamma A 1 ) Gamma1 B 1 C T 2 (sE 2 Gamma A 2 ) Gamma1 B 2 = D ....

....T 2 (A Gamma1 2 E 2 ) i A Gamma1 2 B 2 C T 1 (sI Gamma E Gamma1 1 A 1 ) Gamma1 E Gamma1 1 B 1 : which is in the form of (1.2) or (1.3) if E 1 and A 1 are empty. In doing so, the key step is the equation (2.1) We now give the detail. 3 3 Van Dooren s Algorithm Van Dooren [7] presented a generalization of Kublanovskaya s algorithm [6, 5] for the determination of the Jordan structure of a matrix to determine the structure of a matrix pencils. It is worth mentioning Van Dooren s algorithm works equally well on singular pencils. What we need here is to use Van Dooren s ....

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P. Van Dooren, The computation of Kronecker's canonical form of a singular pencil, Linear Algebra and Its Applications, 27 (1979), pp. 103--140.

....Engineering Sciences under contract TFR 222 97 112. 24 BOUNDS FOR THE DISTANCE BETWEEN 25 unitary similarity transformations. This milestone contribution has stimulated several subsequent papers on algorithms for the numerical computation of Jordan and Kronecker structure information (e.g. see [40, 29, 28, 41, 26, 3, 11, 12, 30]) including several important contributions by herself (e.g. see [34, 35, 31, 32, 4, 36, 37] Morever, new insight and understanding of the mathematical theory of orbits and bundles of matrices and matrix pencils have lead to new results, which in turn stimulate the development of improved ....

....looks quite complicated in the general case, but most matrix pencils have a more simple Kronecker structure. Almost all rectangular m Theta n pencils A Gamma B (m 6= n) have the same KCF, depending only on m and n and this KCF only includes L j (if m n) and L T j blocks otherwise (e.g. see [41, 10, 14]) This corresponds to the generic case when A Gamma B has full rank for any scalar . It follows that generic rectangular pencils have no regular part. We remark that any pencil with only L j or only L T j blocks has full rank, but it is only one of them which is the generic structure. Square ....

P. Van Dooren, The computation of Kronecker's canonical form of a singular pencil. Lin. Alg. Appl., 27 (1979), 103--141.

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P. Van Dooren, The computation of Kronecker's canonical form of a singular pencil, Linear Algebra Appl., 27 (1979), pp. 103--140.

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P. Van Dooren. The computation of Kronecker's canonical form of a singular pencil. Lin. Alg. & Appl., 27:103--141, 1979.

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P. Van Dooren. The computation of Kronecker's canonical form of a singular pencil. Lin. Alg. & Appl., 27:103--141, 1979.

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Van Dooren P., "The computation of Kronecker`s canonical form of a singular pencil", Lin. Alg. & Appl., 27, 103--141, (1979).

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P. Van Dooren. The computation of Kronecker's canonical form a singular pencil. Lin. Alg. & Appl., 27:103--141, 1979.

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Van Dooren P., The Computation of Kronecker's Canonical Form of a Singular Pencil. Linear Algebra and Its Applications, Vol. 27, pp. 103--141, 1979.

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P. Van Dooren, The computation of Kronecker's canonical form of a singular pencil, Linear Algebra and Its Applications, 27 (1979), pp. 103--140.

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P.Van Dooren. The computation of Kronecker's canonical form of a singular pencil. Lin. Alg. Appl., 27:103#141, 1979.

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