I am looking for general solution of the form B=XAX−1 with the following constraintsX−1=X′where X is unknown matrix and A,B,X, are 3×3 matrices.The problem appears peculiar and thus I wish to simplify it further to more practical problem X is rotation matrix (R), and A is plane equation matrix, B is resulted homography in the sense ofB=XAX−1=XAXT=R(I−tnTd)RT,where t is translation vector, n is plane normal vector, and d is signed distance, all three are knownMy dirty solution so far is as follows : Finding the Jordan Canonical form:MBM−1=J=NAN−1and then X is obviously X=N−1Mbut this works only if the canonical forms are the same, otherwise cannot be utilized for solution.The second dirty trick is to solve a system of Linear equations in terms ofAX=BXbut i cannot figure out how to program this in the software (e.g. Matlab linsolve). Could you help me to clarify how to write system of linear equations in the shape of 9 rows?The trick is how to notate them on the "right side" to use in ready to use functions for solving SLE.

Question

I am looking for general solution of the form B=XAX−1 with the following constraintsX−1=X′where X is unknown matrix and A,B,X, are 3×3 matrices.The problem appears peculiar and thus I wish to simplify it further to more practical problem X is rotation matrix (R), and A is plane equation matrix, B is resulted homography in the sense ofB=XAX−1=XAXT=R(I−tnTd)RT,where t is translation vector, n is plane normal vector, and d is signed distance, all three are knownMy dirty solution so far is as follows : Finding the Jordan Canonical form:MBM−1=J=NAN−1and then X is obviously X=N−1Mbut this works only if the canonical forms are the same, otherwise cannot be utilized for solution.The second dirty trick is to solve a system of Linear equations in terms ofAX=BXbut i cannot figure out how to program this in the software (e.g. Matlab linsolve). Could you help me to clarify how to write system of linear equations in the shape of 9 rows?The trick is how to notate them on the &quot;right side&quot; to use in ready to use functions for solving SLE.

benehmenshgf · Accepted Answer

Case 1. A,B are not orthogonally similar; no solutions.    Case 2. A,B are orthogonally similar; A=PBPT, where P∈O(n). Let C(A)={X∈Mn(R);AX=XA}. Then, it is not difficult to show    Proposition {X∈O(n);A=XBXT}=(C(A)∩O(n))P.    It remains to obtain C(A)∩O(n). When n=3, C(A) is a vector space of dimension 3,5 or 9 (when A is a scalar matrix).    Generically (randomly choose A), A is cyclic and C(A)={aI+bA+cA2;a,b,c∈R}. Generically, the eigenspaces of A are not orthogonal and, consequently, C(A)∩O(n)=±In and X=±P.    EDIT. A,B are orthogonally similar iff the couples (A,AT) and (B,BT) are similar. Then, practically, one studies the linear system    (S)AX=XB,ATX=XBT.    If A,B are not orthog. similar, then the sole solution is X=0.    If A,B are orthog. similar, then, for a generic A, (S) admits a vector space of solutions of dimension 1: X=uX0. It remains to calculate u so that ||uX0||2=1.

I am looking for general solution of the form B=XAX^{-1} with the following constraints

Answered question

Answer & Explanation

New Questions in Algebra I