© Hindawi Publishing Corp. EXPRESSION FOR A GENERAL ELEMENT OF AN SO(n) MATRIX

We derive the expression for a general element of an SO(n) matrix. All elements are obtained from a single element of the matrix. This has applications in recently developed methods for computing Lyapunov exponents.


Introduction.
Matrix representations of the SO(n) group have played an important role in mathematical physics [5, 6].They continue to be used in many fields to this day [4, 7, 8].They also play a crucial role in new methods for computing Lyapunov exponents [2, 3].
In this paper, we obtain the expression for a general element of an SO(n) matrix Q (n) for n ≥ 3.This offers significant advantages in generalizing the recent Lyapunov spectrum calculation methods [2, 3] to higher dimensions.We demonstrate that expressions for all elements can be obtained from the expression of a single matrix element by suitable operations.As an example of the application of these results, we derive the elements of an SO(3) matrix in Section 3. The standard expressions are obtained as expected.SO(n) matrix.In this section, we derive the expression for a general element of an SO(n) matrix denoted by Q (n) (for n ≥ 3).In all the expressions below, it is implicitly assumed that n ≥ 3.

General element of an
We start by deriving the expression for the element 1n .Then we prove that all other elements of Q (n) can be obtained from this single element and give explicit expressions for these elements.This method is based on the representation of the group SO(n) as a product of n(n−1)/2 n×n matrices, which are simple rotations in the (i − j)th coordinates [1].Proposition 2.1.An SO(n) matrix Q (n) can be represented as the following product of simple rotations (see [1]): (1,2) O (1,3) where O (i,j) is given as where r = (i − 1)(2n − i)/2 + j − i.
1n is given by the expression (2.13) Next, we prove that all other elements of Q (n) can be obtained from the single element Q (n) 1n (derived above).To show this, we need some preliminary results contained in Lemmas 2.3 and 2.4 proved below.
We are now in a position to prove that we can obtain all rows of Q (n) given only the first row.
We next prove a result analogous to Lemma 2.5, but for columns instead of rows.Combining Lemmas 2.5 and 2.6 will give us the desired result of obtaining all elements of Q (n) from a single element.Lemma 2.6.For n ≥ 3, given the nth column of Q (n) , the (n − 1)th column is given by the following expression: , i= 1, 2,...,n. (2.40) The other columns are given by where The proof of this lemma is by induction and is straightforward (though laborious).So we omit the proof.
Lemma 2.6 implies that given the last column of Q (n) , we can derive the other columns.In particular, given 2), we can obtain the first row.Once the first row is known, using Lemma 2.5, all other rows can be derived.Therefore, we see that from one element of Q (n) , namely, Q (n) 1n we can generate the whole SO(n) matrix by performing suitable operations.Thus we have proved the following theorem.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation