Canonical Forms, Basic Definitions and Properties
In this chapter, linear algebra is employed for the study of symmetry, followed by graph-theoretical interpretations. The application of the methods presented in this chapter is not limited to geometric symmetry. Thus, the symmetry studied here can more a
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Canonical Forms, Basic Definitions and Properties
4.1
Introduction
In this chapter, linear algebra is employed for the study of symmetry, followed by graph-theoretical interpretations. The application of the methods presented in this chapter is not limited to geometric symmetry. Thus, the symmetry studied here can more appropriately be considered as topological symmetry. The methods considered in this chapter can be considered as special techniques for transforming the matrices into block triangular forms. These forms allow good saving of computation effort for many important problems such as computing determinants, eigenvalue problems and solution of linear system of equations. For each of these tasks with dimension N, the computing cost grows approximately with N3. Therefore, reducing, for example, the dimension to N/2, the effort decreases eight times which is a great advantage. Here different canonical forms are presented. Methods are developed for decomposing and healing of the graph models associated with these forms for efficient calculation of the eigenvalues of matrices associated with these forms [1–5]. The formation of divisors and co-divisors, using a graph-theoretical approach, is developed by Rempel and Schwolow [6] and well described in reference [7]. Here, only symmetric forms are presented, since simple graph-theoretical concepts are sufficient for their formation. Two important forms known as tridiagonal and penta-diagonal forms are also presented, and methods are provided for their decomposition. It is shown that different canonical forms can be derived from the block tri-diagonal form [8, 9].
4.2
Decomposition of Matrices to Special Forms
In this section, a 2N 2N symmetric matrix M is considered with all entries being real. For four canonical forms, the eigenvalues of M are obtained using the properties of its submatrices. A. Kaveh, Optimal Analysis of Structures by Concepts of Symmetry and Regularity, DOI 10.1007/978-3-7091-1565-7_4, © Springer-Verlag Wien 2013
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4.2.1
4 Canonical Forms, Basic Definitions and Properties
Canonical Form I
In this case, matrix M has the following pattern: 2 M¼4
ANN
0NN
0NN
ANN
3 5
(4.1) 2N2N
Considering the set of eigenvalues of the submatrix A as fλAg , the set of eigenvalues of M can be obtained as follows: fλMg ¼ fλAg[fλAg:
(4.2)
where the sign [ is used for the collection of the eigenvalues of the submatrices and not necessarily their union. Since det M ¼ det A det A, the proof becomes evident. Here ‘det’ stands for the determinant. Form I can be generalised to a decomposed form with diagonal submatrices A1, A2, A3, . . ., Ap of different dimensions, and the eigenvalues can be calculated as follows: fλMg ¼ fλA1 g[fλA2 g[fλA3 g[ . . . [fλAp g:
(4.3)
The proof follows from the fact that det M ¼ det A1 det A2 det A3 . . . det Ap. Example 4.1. Consider the matrix M as follows: 2
1 63 M¼4 0 0
2 4 0 0
0 0 1 3
3 0 07 ; 25 4
1 2 with A ¼ . 3 4 Since {λA} ¼ {0.3723, 5.3723}, therefore {λM} ¼ {0.3723, 5.3723, 0.3
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