Let F be a field.
Consider U, the set of n times n strictly upper triangular matrices in F.
For X, Y in U, we call them similar if there exists some S, which is non-degenerate and upper triangular, such that
X= SYS-1
My question is, how to describe U/~, ~ being the similairy relation defined as above.
I guess there should be some standard/cannonical matrices in U that classify all others.
And then I am wondering the algorithm --- how to find the equivalence class of X, or the standard form of X?
Any answer, reference, or hints are welcome.
Dear Zhuo:
If i understand your question will, then:
Let U:=Un(F) be the algebra of upper triangular n×n matrices over a field F. there a classification of all simple U-modules (up to isomorphism).
The Jacobson radical J(R) of a ring R consists of all elements which act by zero in all simple left R-modules. It follows that the simple modules of R are naturally identified with the simple modules of R/J(R).
The Jacobson radical has an important alternate characterization as consisting of all elements r such that 1−xr is invertible for all x. Using this characterization, I claim that the Jacobson radical of R=Un(F) consists precisely of the strictly upper triangular matrices. The quotient R/J(R) is isomorphic to Fn, where the isomorphism sends an upper triangular matrix to its diagonal, and the simple modules of Fn are identified with the n copies of F.
Un(F) happens to be a quiver algebra of a finite acyclic quiver, and a similar statement is true in this generality: the simple modules of such an algebra are naturally in bijection with the vertices of the corresponding quiver.
Continue ...
Let us take as an example this hypothesis:
Let T1,T2∈C^n×n be upper triangular matrices. If there exists a nonsingular matrix P such that T1=PT2P^−1, then there is a nonsingular uppper triangular matrix T such that T1=TT2T^−1.
It is false for the following reason. The diagonal elements of a triangular matrix are its eigenvalue. If they are pairwise distinct, the matrix is similar to its diagonal.
Assume now that two upper triangular matrices have the same diagonal elements, pairwise distinct, but not in the same order. Then they are similar. Yet there does not exist an upper triangular P such that T2=PT1P^−1, because conjugating by an upper triangular matrix does not change the diagonal elements.
This is related to the classification of flags.
Also:
Let upper triangular matrix T be also a differential T^2=0. Then, acting by upper triangular conjugation one can get a matrix with at most one 1 in each row and each column, all other elements equal to zero. Moreover, such a matrix (normal form) is unique in the upper triangular conjugacy class. But from the point of view of all conjugations two such a differentials are equivalent if and only if the dimensions of its homologies coincides.
Finally note:
Such differentials and an action of the upper-triangular group naturally arise in a context of Morse theory. A strong Morse function and a generic Riemannian metric on a closed manifold generates Morse complex with an upper triangular Morse differential (upper triangular with respect to the ordering by critical value). A change of a metric reflects as an upper-triangular conjugation of Morse differential. The statement above (on the unicity of normal form in a case of field coefficients) was formulated and proved by Barannikov and attributed to J.Cerf (S. A. Barannikov, “The framed Morse complex and its invariants”, Adv. Soviet Math. 21. (1994), 93–115.).
I hope this answer will be helpful ....
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