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By A. Parshin,I. Shafarevich

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Thus A is pure imaginary and we get that x(t) is periodic with period 2-7r/|A|. Symmetric tridiagonal matrices have particularly fast and efficient eigenvalue algorithms. Later sections deal with the cases of nonunit masses m^ and nonzero damping constants bi. 3 Generalized Hermitian Eigenproblems J. Demmel We assume that A and B are n by n Hermitian matrices and that B is positive definite. We call A — XB a definite matrix pencil, or definite pencil for short. Here A is a variable Chapter 2. 1 For convenience we will refer to eigenvalues, eigenvectors, and other properties of the pencil A — XB.

If we take a subset of k columns of X (say X = X ( : , [2,3,5]) = columns 2, 3, and 5), then these columns span an eigenspace of A — XB. If we take the corresponding submatrix AA = diag(A^>22, AA,SS, A^i55) of AA, and similarly define AB, then we can write the corresponding partial eigendecomposition as X*AX = A A and X*BX = A^. If the columns in X are replaced by k different vectors spanning the same eigensubspace, then we get a different partial eigendecomposition, where A^ and AB are replaced by different k-by-k matrices AA and AB such that the eigenvalues of the pencil AA — XAs are those of A A — A AS, though the pencil AA — XAB may not be diagonal.

Again, an eigenspace spanned by the eigenvectors of a cluster of eigenvalues may be much better conditioned than the individual eigenvectors. 7 for further details. 6 for the Hermitian eigenproblem. 1. Compute all the eigenvalues to some specified accuracy. 2. , n}, including the special cases of the largest m eigenvalues A n _ m+ i through Xn, and the smallest m eigenvalues AI through A m . Again, the desired accuracy may be specified. 3. ]. Again, the desired accuracy may be specified. Chapter 2.