a matrix similarity problem. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:11:55Z http://mathoverflow.net/feeds/question/49679 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49679/a-matrix-similarity-problem a matrix similarity problem. Sunni 2010-12-16T19:49:39Z 2010-12-16T22:35:01Z <p>I'd like to know whether the following statement is true or not.</p> <p>Let $T_1, T_2\in \mathbb{C}^{n\times n}$ be upper triangular matrices. If there exists a nonsingular matrix $P$ such that $T_1=PT_2P^{-1}$, then there is a nonsingular uppper triangular matrix $T$ such that $T_1=TT_2T^{-1}$. </p> http://mathoverflow.net/questions/49679/a-matrix-similarity-problem/49680#49680 Answer by Petya for a matrix similarity problem. Petya 2010-12-16T19:56:08Z 2010-12-16T22:35:01Z <p>It is not true. </p> <p>Let upper triangular matrix $T$ be also a differential, i.e. $T^2=0$. 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. </p> <p>The simplest example through differentials are matrices</p> <p>001</p> <p>000</p> <p>000</p> <p>and</p> <p>000</p> <p>001</p> <p>000</p> <p>They are not in the same upper triangular conjugacy class.</p> <p>Remark: 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.).</p> http://mathoverflow.net/questions/49679/a-matrix-similarity-problem/49682#49682 Answer by Denis Serre for a matrix similarity problem. Denis Serre 2010-12-16T20:41:32Z 2010-12-16T20:41:32Z <p>It is <strong>false</strong> for the following obvious reason. The diagonal elements of a triangular matrix are its eigenvalue. If they are pairwise distinct, the matrix is similar to its diagonal. </p> <p>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 $T_2=PT_1P^{-1}$, because conjugating by an upper triangular matrix does not change the diagonal elements.</p> <p>This is related to the classification of flags.</p>