Help with Matrix PQ=QR - Find a Simple Answer

[Vicis]

There must be a simple answer to this problem but ill be damed if i can find it.

I need to find matrix P,Q,R So that PQ=QR

Ive tried so many times but i can't solve it, I've been to three different applicable maths books looking for help but they all where dead ends.

Could someone please help me.

 

[christianjb]

Which matrices if any are given?

 

[Hurkyl]

One solution is for P, Q, and R to each be a zero matrix with the appropriate dimensions.

 

[christianjb]

Well, if P,Q,R are the identity matrix then that would work, but at the moment it's not a well posed problem.

 

[srikanth.isro]

1)P and R can be any matrix of the same order as that of Q and Q must be a null matrix.
2)All the three should be identity matrices of the same order.

 

[daniel_i_l]

There're an infinite amount of solutions to this problem.
In addition to everything said above the equation will hold if all three matricies are powers of some matrix, or if they're scalar matricies.

 

[radou]

Assume Q is regular, and P, R are square matrices. If you multiply PQ = QR from the left with Q^-1, you obtain Q^-1 P Q = R, which implies that R is similar to P.

Further on, if R is similar to P, then there exists a regular matrix Q such that R = Q^-1 P Q. Multiply from the left with Q and obtain QR = PQ.

So, PQ = QR <=> R is similar to P. That could be one point of view. An example of similar matrices are matrix representations of linear operators in different basis sets.