AM  Vol.4 No.3 , March 2013
Derivative of a Determinant with Respect to an Eigenvalue in the LDU Decomposition of a Non-Symmetric Matrix
ABSTRACT

We demonstrate that, when computing the LDU decomposition (a typical example of a direct solution method), it is possible to obtain the derivative of a determinant with respect to an eigenvalue of a non-symmetric matrix. Our proposed method augments an LDU decomposition program with an additional routine to obtain a program for easily evaluating the derivative of a determinant with respect to an eigenvalue. The proposed method follows simply from the process of solving simultaneous linear equations and is particularly effective for band matrices, for which memory requirements are significantly reduced compared to those for dense matrices. We discuss the theory underlying our proposed method and present detailed algorithms for implementing it.


Cite this paper
M. Kashiwagi, "Derivative of a Determinant with Respect to an Eigenvalue in the LDU Decomposition of a Non-Symmetric Matrix," Applied Mathematics, Vol. 4 No. 3, 2013, pp. 464-468. doi: 10.4236/am.2013.43069.
References
[1]   M. Kashiwagi, “An Eigensolution Method for Solving the Largest or Smallest Eigenpair by the Conjugate Gradient Method,” Transactions of the Japan Society for Aeronautical and Space Sciences, Vol. 1, 1999, pp. 1-5.

[2]   M. Kashiwagi, “A Method for Determining Intermediate Eigensolution of Sparse and Symmetric Matrices by the Double Shifted Inverse Power Method,” Transactions of JSIAM, Vol. 19, No. 3, 2009, pp. 23-38.

[3]   M. Kashiwagi, “A Method for Determining Eigensolutions of Large, Sparse, Symmetric Matrices by the Preconditioned Conjugate Gradient Method in the Generalized Eigenvalue Problem,” Journal of Structural Engineering, Vol. 73, No. 629, 2008, pp. 1209-1217.

[4]   M. Kashiwagi, “A Method for Determining Intermediate Eigenpairs of Sparse Symmetric Matrices by Lanczos and Shifted Inverse Power Method in Generalized Eigenvalue Problems,” Journal of Structural and Construction Engineering, Vol. 76, No. 664, 2011, pp. 1181-1188. doi:10.3130/aijs.76.1181

 
 
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