Summary. An algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eigensystem. The explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues are discussed. The sensitivity of the computed eigenvectors to errors in the updated eigenvalues is shown by a perturbation analysis.

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Several decompositions of, symmetric matrices for calculating inertia and solving systems of linear equations are discussed. New partial pivoting strategies for decomposing symmetric matrices are introduced and analyzed.

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Abstract. The fast matrix multiplication algorithm by Strassen is used to obtain the triangular factorization of a permutation of any nonsingular matrix of ordern in 2.35, i.e. if n > (2.35)5 « 100.Strassen uses block LDU factorization (Householder [2, p. 126]) recursively to compute the inverse of a matrix of order «22* by mlk divisions, g (6/5)m37* -mlk multiplications, and ;£ (6/5)(5 + m)m2lk -7(m2k)2 additions. The inverse of a matrix of order n could then be computed by ^ (5.64)«logi'7 arithmetic operations.