By Peter Lancaster
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Extra resources for Lambda-matrices and vibrating systems
Since the latent vectors are obtained from these by the non-singular -1/2, it follows that the latent transformation represented by A vectors are also linearly independent. This completes the theorem. We remarked in the proof that the latent roots of the pencil coincide with the eigenvalues of the real symmetric matrix —B, and the eigenvalues are known to be real. We have therefore: COROLLARY Under the assumptions o f the theorem all the latent roots and latent vectors of the pencil are real. 11 could perhaps be proved under the weaker condition that A is merely non-singular, rather than being positive definite, for A -1/ 2 would still exist though it would no longer necessarily be real.
2) as our definition for R we deduce that a simple matrix pencil of order n has n linearly independent left latent vectors. 1), the latent vectors are said to satisfy a biorthogonality condition with respect to the matrix A. 3) where d,, is the Kronecker delta. The choice of the latent vectors (corresponding to multiple roots) which gives rise to this bierthogonality property warrants further examination. LEMMA A simple matrix pencil has x linearly independent right (or left) latent vectors associated with a latent root of multiplicity x.
Thus, the existence of matrices which are not of simple structure (or which are defective) implies the existence of real, regular, symmetric pencils which are not simple. 7 GEOMETRIC IMPLICATIONS OF THE JORDAN CANONICAL FORM Let A be a matrix which does not have simple structure. 12 To an eigenvalue l. of A for which the multiplicity exceeds the index there corresponds at least one right (left) eigenvector a such that y'ii = 0 for every vector y in the subspace of left (right) eigenvectors of l.