#Math101 Session 16
Time：May 30, 2022, 13:30--15:30
Location：447 Qi Xiang Building, Chang'an Campus
Speaker：Mr. Sun, Li (Class of 2025)
Title：On the Sylvester equationAX-XB=C
Abstract：Equations of the formAX-XB=Care called Sylvester equations, whereA, B, Care complex matrices of appropriate dimensions. This is a linear system of equations with various applications in theoretical and applied mathematics. It is uniquely solvable for any givenCif and only if the corresponding homogeneous equationAX-XB=0admits only the trivial solutionX=0. In this talk, we will first show that this latter condition is in turn equivalent toAandBsharing no eigenvalue. Secondly, given complex matricesAandBof the same order, sayn, one defines the linear transformationTon the space of order-ncomplex matrices byT(X)=AX-XB. The eigenvalues ofTand the condition whenTis diagonalizablewillbe determined.