characteristic matrix of diagonal element cross-section
Denote by the set of all matrices over . Let be the function which extracts the th diagonal element of a matrix. Finally denote by the set .
A diagonal element of an upper triangular matrix is of course an eigenvalue of that matrix, so the particular that one plugs into this lemma is typically either a sequence of eigenvalues for the given matrices, or a sequence of values that one thinks may be eigenvalues for these matrices. The “cross-section” in the title refers to that there is one for each matrix .
The result gets more interesting if one knows something more about than what was explicitly required above. A particular example is that if the given matrices commute then will be a commutative (http://planetmath.org/Commutative) algebra and consequently will commute with all of the given matrices as well.
Let be the set of those row indices for which should be , i.e.,
and for each let be such that . Define
for all and let . Since all the matrices involved are upper triangular, a diagonal element in is simply the product of the corresponding diagonal elements in all of . If then and thus for . If instead then
This is by definition of nonzero, and since it is independent of it follows that it is the only nonzero value of a diagonal element of . Hence the wanted . ∎
|Title||characteristic matrix of diagonal element cross-section|
|Date of creation||2013-03-22 15:29:45|
|Last modified on||2013-03-22 15:29:45|
|Last modified by||lars_h (9802)|