Page 258 - Advanced Linear Algebra

242 Advanced Linear Algebra

Unitary/orthogonal matrices play the role of change of basis matrices when we
restrict attention to orthonormal bases. Let us first note that if 8 ~²" Á Ã Á " ³

is an ordered orthonormal basis and

# ~ " bÄb "

$ ~ " bÄb "

then
º#Á $» ~ bÄb ~ ´#µ h ´$µ 8
8

where the right hand side is the standard inner product in - and so ž # $ if
. We can now state the analog of Theorem 2.9.
8
and only if ´#µ ž ´$µ 8
Theorem 10.14 If we are given any two of the following:
)
(
1 A unitary/orthogonal d matrix ,
)
8
2 An ordered orthonormal basis for - ,
)
9
3 An ordered orthonormal basis for - ,
then the third is uniquely determined by the equation
(~ 4 89Á
=
Proof. Let ~¸ ¹ be a basis for . If is an orthonormal basis for , then
=
8
9

º Á » ~ ´ µ h ´ µ 9 9

(
where ´ µ is the th column of ( ~ 4 Á 9 8 9 . Hence, is unitary if and only if 8
is orthonormal. We leave the rest of the proof to the reader.
Unitary Similarity
We have seen that the change of basis formula for operators is given by
´µ ~ 7´µ 7 c

Z
8
8
where is an invertible matrix. What happens when the bases are orthonormal?
7
Definition
(
)
1 Two complex matrices ( and ) are unitarily similar also called
unitarily equivalent) if there exists a unitary matrix for which
<
) ~ <(< c ~ <(< i
The equivalence classes associated with unitary similarity are called
unitary similarity classes.
2 Similarly, two real matrices and are orthogonally similar also called
(
)
)
(
orthogonally equivalent) if there exists an orthogonal matrix for which
6
) ~ 6(6 c ~ 6(6 !
The equivalence classes associated with orthogonal similarity are called
orthogonal similarity classes.