If and is supertriangular then
theorem: Let be commutative ring with identity.
If an n-square matrix
is supertriangular then .
proof: Find the characteristic polynomial of by computing the determinant
of . The square matrix
is a triangular matrix
. The determinant of a triangular matrix is the product of the diagonal element of the matrix. Therefore the characteristic polynomial is and by the Cayley-Hamilton theorem
the matrix satisfies the polynomial
. That is .
QED
Title | If and is supertriangular then |
---|---|
Canonical name | IfAinMnRAndAIsSupertriangularThenAn0 |
Date of creation | 2013-03-22 13:44:39 |
Last modified on | 2013-03-22 13:44:39 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 12 |
Author | Daume (40) |
Entry type | Theorem |
Classification | msc 15-00 |