Find the equivalance of AB and BC

[onako]

Suppose a (n x 3) matrix B is given, n>3. Also, suppose a matrix (3 x 3) matrix C is given, whose columns correspond to eigenvectors of X'LX, for some symmetric, real L. How could I state the right multiplication of B, ie., BC, is equivalent to left multiplication of the form AB, for some (n x n) matrix A?

 

[chiro]

Suppose a (n x 3) matrix B is given, n>3. Also, suppose a matrix (3 x 3) matrix C is given, whose columns correspond to eigenvectors of X'LX, for some symmetric, real L. How could I state the right multiplication of B, ie., BC, is equivalent to left multiplication of the form AB, for some (n x n) matrix A?

Hey onako.

Have you tried expanding out the form of the matrix using your A,B, and C matrices?

Although the matrices have the same size left and right (AB = BC), the system does not have a proper basis which means that you can't use these kinds of methods (as far as I know) (unless n = 3). A basis requires n linearly independent vectors in your basis matrix.

If you just want to know how to write the above without any specific details you can write AB = BC. If you want to add more detail you're probably going to have to expand out the system in terms of summations for each element on the LHS and RHS and then do what you have to do.