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1959: The pseudoinverse of a rectangular or singular matrix and its application to the solution of systems of linear equations. SIAM Review, 1, 38–43. 9. T. F. 1992: A Generalised Complex Symmetric Engensolver. Computers & Structures, 43(6), 1183–1186. 10. M. 1991: Fundamentals of Singular Value Decomposition. Proceedings of the 9th International Modal Analysis Conference, Firenze, Italy, 1515–1521. 11. Z. 1976: Generalised Inverses and Applications. , New York, USA. 12. Ogata, K. 1990: Modern Control Engineering.
The LR algorithm requires that [A] be decomposed first as a product of lower- and upper-triangular matrices [L] and [R] (the same as the LU decomposition). Then, a new matrix is created by exchanging the two triangular matrices to form [R][L]. The decomposition is again performed on [R][L]. The repetition of this process will eventually lead to a product of a unit matrix and an upper-triangular matrix whose diagonal elements are the eigenvalues of the original [A]. It is known that every operation brings about a similar matrix.
38) The inverse of [A] can be easily derived as: [ A ] –1 = [ M ] n [ M ] n–1 . . 40) where [U] is an upper triangular matrix. 34). 2 Inverse of a square complex matrix A square complex matrix [C] can be inverted using the inverse of a real matrix. The idea is to transform the inverse into that of a real matrix with double-sized order. 46) Alternatively, matrices [X] and [Y] can be determined directly from the definition of matrix inverse. 48) should produce the same results. 3 Pseudo inverse of a matrix When a matrix is either non-square, or rank-deficient or both, normal inverse does not apply anymore.