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Тезисы докладов

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itle{Bounds for interior eigenvalues of Hermitian matrices}

author{L.~Yu.~Kolotilina}

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As is well known, the convegence of iterative methods for solving linear equations depends on the distribution of the eigenvalues of the coefficient matrix. In this paper, we describe the best possible distribution of all the eigenvalues of a Hermitian matrix presented in block $2 imes2$ form. More precisely, let an $n imes n$ Hermitian matrix $A=left[egin{array}{cc} A_{11}&A_{12}cr A^*_{12}&A_{22} end{array} ight]$ be positive definite and assume that $A_{12} e0$. Denote

$$ R=A^{-1/2}_{11}A_{12}A^{-1/2}_{22}, $$

$$ mu^{(i)}_pm(A)=frac{sigma_i(A_{12})}{sigma_i(R)}pmsigma_i(A_{12}), i=1,ldots,rank A_{12}, $$

where $sigma_i$ are nonincreasingly ordered singular values. We prove the following results.

Theorem 1. If for some $k$, $0le klambda_i(A)=mu^{(i)}_+(A), i=1,ldots,k, $$

then $$

lambda_{k+1}(A)gemu^{(k+1)}_+(A). $$

Theorem 2. If for some $k$, $0le klambda_{n-i+1}(A)=mu^{(i)}_-(A), i=1,ldots,k, $$

then $$

lambda_{n-k}(A)lemu^{(k+1)}_-(A). $$

An important consequence of the results obtained is that, for any nonsingular matrix $D=left[ egin{array}{cc} D_1&0 0&D_2end{array} ight]$, from the conditions $$

frac{lambda_i(D^*AD)}{lambda_{n-i+1}(D^*AD)}%& = frac{lambda_i(D^{-1/2}_A A D^{-1/2}_A)}{lambda_{n-i+1}(D^{-1/2}_A A D^{-1/2}_A)}, i=1,ldots,k, $$

it follows that $$

frac{lambda_{k+1}(D^*AD)}{lambda_{n-k}(D^*AD)}%&

Примечание. Тезисы докладов публикуются в авторской редакции

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