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 .
Lemma.
Let be a field.
Let a sequence of upper
triangular matrices be given, and denote by the unital algebra generated by these matrices.
For every sequence of
scalars there exists a matrix such that
for all .
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 βcross-sectionβ 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.
Proof.
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 . β
Title | characteristic matrix of diagonal element cross-section |
---|---|
Canonical name | CharacteristicMatrixOfDiagonalElementCrosssection |
Date of creation | 2013-03-22 15:29:45 |
Last modified on | 2013-03-22 15:29:45 |
Owner | lars_h (9802) |
Last modified by | lars_h (9802) |
Numerical id | 4 |
Author | lars_h (9802) |
Entry type | Theorem |
Classification | msc 15A30 |