Mathematical Methods for Physicists: A concise introduction - Site Map

SIMILARITY TRANSFORMATION
where ~ S is a non-singular matrix, the transition matrix from the new coordinate
system to the old system. With these, Eq. (3.51a) becomes
or
Combining this with Eq. (3.51) gives
~SR 0 ˆ ~A ~Sr 0
R 0 ˆ ~S 1 ~A ~Sr 0 :
~A 0 ˆ ~S 1 ~ A ~ S; …3:52†
where ~A 0 and ~A are similar matrices. Eq. (3.52) is called a similarity transformation.
Generalization of this idea to n-dimensional vectors is straightforward. In this
case, we take r and R as two n-dimensional vectors in a particular basis, having
their coordinates connected by the matrix ~A (a n n square matrix) through Eq.
(3.51a). In another basis they are connected by Eq. (3.51b). The relationship
between ~A and ~A 0 is given by Eq. (3.52). The transformation of ~A into ~S 1 ~A ~S
is called a similarity transformation.
All identities involving vectors and matrices will remain invariant under a
similarity transformation since this arises only in connection with a change in
basis. That this is so can be seen in the following two simple examples.
Example 3.10
Given the matrix equation ~A ~B ˆ ~C, and the matrices ~A, ~B, ~C subjected to the
same similarity transformation, show that the matrix equation is invariant.
Solution: Since the three matrices are all subjected to the same similarity transformation,
we have
~A 0 ˆ ~S A ~ S ~ 1 ; ~B 0 ˆ ~S ~B S ~ 1 ; C ~ 0 ˆ ~S C ~ S ~ 1
and it follows that
~A 0 ~B 0 ˆ…~S ~A ~S 1 †… ~S ~B ~S 1 †ˆ ~S ~A~I ~B ~S 1 ˆ ~S ~A ~B ~S 1 ˆ ~S ~C ~S 1 ˆ ~C 0 :
Example 3.11
Show that the relation ~AR ˆ ~Br is invariant under a similarity transformation.
Solution: Since matrices A ~ and ~B are subjected to the same similarity transformation,
we have
~A 0 ˆ ~S A ~ S ~ 1 ; ~B 0 ˆ ~S ~B S ~ 1
we also have
R 0 ˆ ~SR;
r 0 ˆ ~Sr:
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