For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a lower triangular with an upper triangular matrix is not necessarily triangular either.
32.
If the entries on the main diagonal of a ( upper or lower ) triangular matrix are all 1, the matrix is called ( upper or lower ) "'unitriangular " '.
33.
A more precise statement is given by the Jordan normal form theorem, which states that in this situation, " A " is similar to an upper triangular matrix of a very particular form.
34.
A central series is analogous in Lie theory to a adjoint action ( more prosaically, a basis in which each element is represented by a strictly upper triangular matrix ); compare Engel's theorem.
35.
Since the determinant of a triangular matrix is the product of its diagonal entries, if " T " is triangular, then Thus the eigenvalues of " T " are its diagonal entries.
36.
For example, we can conveniently require the lower triangular matrix " L " to be a unit triangular matrix ( i . e . set all the entries of its main diagonal to ones ).
37.
For example, we can conveniently require the lower triangular matrix " L " to be a unit triangular matrix ( i . e . set all the entries of its main diagonal to ones ).
38.
An example of a group of symplectic matrices is the group of three symplectic 2x2-matrices consisting in the identity matrix, the upper triagonal matrix and the lower triangular matrix, each with entries 0 and 1.
39.
In fact, a matrix " A " over a field containing all of the eigenvalues of " A " ( for example, any matrix over an algebraically closed field ) is similar to a triangular matrix.
40.
Property 5 says that the determinant on matrices is field, properties 11 and 12 can be used to transform any matrix into a triangular matrix, whose determinant is given by property 6; this is essentially the method of Gaussian elimination.
