Why is lower triangular matrix not mentioned in the Jordan normal form definition

jordan-normal-formlinear algebra

In Wikipedia It is written that

Jordan canonical form is an upper triangualr matrix of a particular form called a Jordan matrix representing a linear operator on a finite dimensional vector space with respect to some basis
enter image description here

My confusion :why is lower triangular matrix not mentioned in the Jordan normal form definition ?

My thinking : I can also construct lower triangular matrix in the same pattern

see the diagram below

enter image description here

Best Answer

Every matrix is simlar to its transponse, so a lower triangular Jordan block would be similar to an upper triangular Jordan block. I recall reading a linear algebra book (I think Serge Lang's) where the Jordan form was lower triangular. It's all a matter of reversing the order of the cyclic (sub-)basis.

Related Question