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Matrix Inversion by Rank Annihilation

David J. Edelblute

1966
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Mathematics of Computation
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1. Remarks. The problem of matrix inversion has been extensively explored and a number of methods have been found to solve this problem. However, no one "best" method has been found and the computer programmer must still choose a method which is suited to his particular needs. The purpose of this paper is to outline a method which allows one to bring the matrix into memory one column at a time and overlap input time with computing time. Since the original matrix need not be stored in memory,
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... tored in memory, the use of memory is also efficient, and the number of computations is as small as any known to the writer. If the original matrix is already in memory, this method can also be used to avoid destroying the original matrix. 2. Theory. The reader may easily verify that if A is a nonsingular matrix, U and V are column vectors, and iA -+-UVT) is nonsingular, then (1) (A + uvr -a--ffff. Householder remarks that if 1 + VTA~~iU = 0 then the matrix i A + UVT) is singular [2]. The repeated use of ( 1 ) to find the inverse of an n X n matrix, B, is known as the method of rank annihilation. To use this method we write (2) B = D+t, UiV,T, <=i where D is a matrix of known inverse. Thus we can define a sequence of matrices ¡d) such that Co = D, Ck = D + Z VtVf, *-> " k = 1, • • • , n. = Ck-x + UkVkT, If all d are nonsingular formula (1) gives a sequence of matrices ¡C,-1}, and CV1 = B~\ Clearly the expansion (2) is not unique. Thus we have the problem of choosing a sequence of [/, and Vi which will produce B"1 with the fewest computations and which will require a minimum amount of memory space in a computer. No optimum solution is known to the writer. The purpose of this paper is to point out some advantages of one extremely simple expansion. Let B be the n X n matrix to be inverted and let the matrix U = B -I, where / is the identity matrix. Partition U by columns so that V = [UjU2 ■ ■ ■ Un\. Let V{ be the ith column vector of the identity matrix. Then B = / + ¿ UíVíT. »=i This is the expansion which we shall discuss.

doi:10.2307/2004280
fatcat:4bamigiwdrg7la5o4fgu4munhu