Question

Hi please give an example and explain using the Matrix Transformations below. Will rate! Thank you.

Topic: Matrix Transformation

Given the matrix transformations T1 and T2, under what real-world conditions will T1 o T2 = T2 o T1?

Answer 1

Let the standard matrices of the transformations T₁ and T₂ be A and B respectively. Then AB and BA are the matrices of the transformations T₂oT₁ and T₁oT₂ respectively. Thus, we have T₁oT₂ = T₂oT₁ if AB = BA.

When A is I_n and B is the nxn zero matrix, then A and B commute i.e. AB = BA. Also, when A and B are both nxn diagonal matrices, then AB = BA.

In general, the matrix products AB and BA are defined, and are equal, only if A and B are square matrices of the same size. Also, we know that if the matrices A{\displaystyle A} and B{\displaystyle B} are both diagonalizable, and if , there exists a similarity matrix {\displaystyle P}P such that {\displaystyle P^{-1}AP}P^-1AP and P^-1BP {\displaystyle P^{-1}BP} are both diagonal, then {\displaystyle A}A and {\displaystyle B}B commute. This can be proved as under:

Let A and B be both nxn matrices and let A= PMP^-1,where M is a nxn diagonal matrix (so that P^-1AP = M) and also let B = PNP^-1,where N is a nxn diagonal matrix (so that P^-1BP =N). Then AB = (PMP^-1)( PNP^-1) = PM(P^-1 P)NP^-1 = PM(I_n)NP^-1=PMNP^-1. Similarly, BA =(PNP^-1)( PMP^-1) = PN(P^-1 P)MP^-1 = PN(I_n)MP^-1 = PNMP^-1.

Now, since M and N are both nxn diagonal matrices, hence MN = NM so that PMNP^-1 = PNMP^-1 i.e. AB = BA.