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## Eigendecompositions

Define and . is called an eigenvector matrix of . Since the are orthogonal unit vectors, we see that ; i.e., is a unitary (orthogonal) matrix. The equalities for may also be written or . The factorization

is called the eigendecomposition of . In other words, is similar to the diagonal matrix , with similarity transformation .

If we take a subset of columns of (say = columns 2, 3, and 5), then these columns span an invariant subspace of . If we take the corresponding submatrix of , then we can write the corresponding partial eigendecomposition as or . If the columns in are replaced by different vectors spanning the same invariant subspace, then we get a different partial eigendecomposition , where is a -by- matrix whose eigenvalues are those of , though may not be diagonal.

Next: Conditioning Up: Hermitian Eigenproblems   J. Previous: Equivalences (Similarities)   Contents   Index
Susan Blackford 2000-11-20