Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
preserves the norm of a vector. To see this, premultiplying Eq. (3.48) by
~Y y …ˆ ~X y ~U y ) and using the condition ~U y ~U ˆ ~I, we obtain
or
~Y y ~Y ˆ ~X y ~U y ~U ~X ˆ ~X y ~X …3:49a†
X n
kˆ1
y k
*y k ˆ Xn
kˆ1
x k
*x k :
…3:49b†
This shows that the norm of a vector remains invariant under a unitary transformation.
If the matrix ~U of transformation happens to be real, then ~U is also an
orthogonal matrix and the transformation (3.48) is an orthogonal transformation,
and Eqs. (3.49) reduce to
~Y T ~Y ˆ ~X T ~X; …3:50a†
as we expected.
X n
kˆ1
y 2 k ˆ Xn
kˆ1
x 2 k;
…3:50b†
Similarity transformation
We now consider a di€erent linear transformation, the similarity transformation
that, we shall see later, is very useful in diagonalization of a matrix. To get the
idea about similarity transformations, we consider vectors r and R in a particular
basis, the coordinate system Ox 1 x 2 x 3 , which are connected by a square matrix ~A:
R ˆ ~Ar:
…3:51a†
Now rotating the coordinate system about the origin O we obtain a new system
Ox1x 0 2x 0 3
0 (a new basis). The vectors r and R have not been a€ected by this
rotation. Their components, however, will have di€erent values in the new system,
and we now have
R 0 ˆ ~A 0 r 0 :
…3:51b†
The matrix ~A 0 in the new (primed) system is called similar to the matrix ~A in the
old (unprimed) system, since they perform same function. Then what is the relationship
between matrices ~ A and ~ A 0 ? This information is given in the form
of coordinate transformation. We learned in the previous section that the components
of a vector in the primed and unprimed systems are connected by a
matrix equation similar to Eq. (3.43). Thus we have
r ˆ ~Sr 0 and R ˆ ~SR 0 ;
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