Page 325 - Matrix Analysis & Applied Linear Algebra

5.6 Unitary and Orthogonal Matrices 321

Characterizations

• The following statements are equivalent to saying that a complex
matrix U n×n is unitary.
U has orthonormal columns.
U has orthonormal rows.
U −1 = U .
∗
Ux = x for every x ∈C n×1 .
2 2
• The following statements are equivalent to saying that a real matrix
P n×n is orthogonal.
P has orthonormal columns.
P has orthonormal rows.
T
P −1 = P .
Px = x for every x ∈ n×1 .
2 2

Example 5.6.1
• The identity matrix I is an orthogonal matrix.
• All permutation matrices (products of elementary interchange matrices) are
orthogonal—recall Exercise 3.9.4.
• The matrix  √ √ √ 
1/ 21/ 3 −1/ 6
√ √ √
P =  −1/ 21/ 3 −1/ 6 
√ √
0 1/ 3 2/ 6
T
T
is an orthogonal matrix because P P = PP = I or, equivalently, because
the columns (and rows) constitute an orthonormal set.
1+i −1+i
∗
∗
• The matrix U = 1 is unitary because U U = UU = I or,
2 1+i 1 − i
equivalently, because the columns (and rows) are an orthonormal set.
• An orthogonal matrix can be considered to be unitary, but a unitary matrix
is generally not orthogonal.
In general, a linear operator T on a vector space V with the property that
n
Tx = x for all x ∈V is called an isometry on V. The isometries on
n
are precisely the orthogonal matrices, and the isometries on C are the unitary
matrices. The term “isometry” has an advantage in that it can be used to treat
the real and complex cases simultaneously, but for clarity we will often revert
back to the more cumbersome “orthogonal” and “unitary” terminology.