Mathematical Methods for Physics and Engineering - Matematica.NET

8.15 CHANGE OF BASIS AND SIMILARITY TRANSFORMATIONSand represent x in this basis by the column matrixx =(x 1 x 2 ··· x n ) T ,having components x i . We now consider how these components change as a resultof a prescribed change of basis. Let us introduce a new basis e ′ i , i =1, 2,...,N,which is related to the old basis bye ′ j =N∑S ij e i , (8.90)i=1the coefficient S ij being the ith component of e ′ j with respect to the old (unprimed)basis. For an arbitrary vector x it follows thatN∑ N∑N∑x = x i e i = x ′ je ′ j =i=1j=1x ′ jj=1 i=1N∑S ij e i .From this we derive the relationship between the components of x in the twocoordinate systems asN∑x i = S ij x ′ j,which we can write in matrix form asj=1x = Sx ′ (8.91)where S is the transformation matrix associated with the change of basis.Furthermore, since the vectors e ′ j are linearly independent, the matrix S isnon-singular and so possesses an inverse S −1 . Multiplying (8.91) on the left byS −1 we findx ′ = S −1 x, (8.92)which relates the components of x in the new basis to those in the old basis.Comparing (8.92) and (8.90) we note that the components of x transform inverselyto the way in which the basis vectors e i themselves transform. This has to be so,as the vector x itself must remain unchanged.We may also find the transformation law for the components of a linearoperator under the same change of basis. Now, the operator equation y = A x(which is basis independent) can be written as a matrix equation in each of thetwo bases asBut, using (8.91), we may rewrite the first equation asy = Ax, y ′ = A ′ x ′ . (8.93)Sy ′ = ASx ′ ⇒ y ′ = S −1 ASx ′ .283