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## Eigendecompositions

Define and . is called an eigenvector matrix of . Since the are unit vectors orthogonal with respect to the inner product induced by , we see that , a nonsingular diagonal matrix. The equalities for may also be written or . Thus is diagonal too. The factorizations

(or and ) are called an eigendecomposition of . In other words, is congruent to the diagonal pencil , with congruence transformation .

If we take a subset of columns of (say = columns 2, 3, and 5), then these columns span an eigenspace of . If we take the corresponding submatrix of , and similarly define , then we can write the corresponding partial eigendecomposition as and . If the columns in are replaced by different vectors spanning the same eigensubspace, then we get a different partial eigendecomposition, where and are replaced by different -by- matrices and such that the eigenvalues of the pencil are those of , though the pencil may not be diagonal.

Next: Conditioning Up: Generalized Hermitian Eigenproblems   Previous: Equivalences (Congruences)   Contents   Index
Susan Blackford 2000-11-20