Linear Algebra Topics - Computer Graphics and Geometric Modeling - page 14

Proof. If we think of the rows of the matrix as vectors, then we get the corres-
pondence by associating to each matrix the basis of R n , which consists of the rows
of the matrix. A similar correspondence is obtained by using the columns of the
matrix.

1.4.9. Theorem. Assume n ≥ 1. Let u 1 , u 2 ,..., u n and v 1 , v 2 ,..., v n be orthonor-
mal bases in a real inner product space V . If

n

 1

v = a u , a ΠR , (1.18)

i ij j ij

j =

then A = (a ij ) is an orthogonal matrix. Conversely, let A = (a ij ) be an orthogonal matrix.
If u 1 , u 2 ,..., u n is an orthonormal basis and if v 1 , v 2 ,..., v n are defined by equation
(1.18), then the v 's will also be an orthonormal basis.

Proof. The theorem follows from the following identities

n n n

=∑= Ê

ˆ

˜ ∑ Ê

ˆ
˜ =

ÂÂÂ

d st vv a u a u a a .

Á Á

s t sj j tj j sj tj

j = 1 j = 1 j = 1

There is a complex analog of an orthogonal real matrix.

An n ¥ n complex matrix A is said to be unitary if A T = A T Definition. A A = I, that

is, the inverse of the matrix is just its conjugate transpose.

Lemma 1.4.7 remains true if we replace the word “orthogonal” with the word
“unitary.” In particular, the unitary matrices form a group like the orthogonal ones.

Definition. The group of nonsingular complex n ¥ n matrices under matrix multi-
plication is called the (complex) linear group and is denoted by GL (n, C ). The subgroup
of unitary n ¥ n matrices is called the unitary group and is denoted by U (n). A unitary
matrix that has determinant +1 is called a special unitary matrix . The subgroup of U (n)
of special unitary n ¥ n matrices is called the special unitary group and is denoted by
SU (n).

The analogs of Theorems 1.4.8 and 1.4.9 hold in the complex case. We omit the
details. See, for example, [Lips68] or [NobD77]. We shall run into orthogonal and
unitary matrices again later in this chapter and in Chapter 2 when we talk about
distance preserving maps or isometries.

1.5 Planes

Next, we define the higher-dimensional linear subspaces of Euclidean space. Certainly
vector subspaces of R n should be such spaces, but “translations” of those should count
also.