Download 5.3 Orthogonal Transformations and Orthogonal Matrices

5.3 Orthogonal Transformations and Orthogonal Matrices
Definition 1 (5.3.1). A linear transformation T : Rn → Rn is called orthogonal if it preserves the length
of vectors:
||T (x)|| = ||x|| , for all x ∈ Rn .
If T (x) = Ax is an orthogonal transformation, we say that A is an orthogonal matrix.
Theorem 2 (5.3.2, orthogonal transformations preserve orthogonality). Let T : Rn → Rn be an
orthogonal linear transformation. If v, w ∈ Rn are orthogonal, then so are T (v), T (w).
Note 3. In fact, orthogonal transformations preserve all angles, not just right angles: the angle between
two nonzero vectors v, w ∈ Rn equals the angle between T (v), T (w). This is a homework problem.
Theorem 4 (orthogonal transformations preserve the dot product). A linear transformation
T : Rn → Rn is orthogonal if and only if T preserves the dot product:
v · w = T (v) · T (w)
for all v, w ∈ Rn .
Theorem 5 (5.3.3, orthogonal matrices and orthonormal bases). An n × n matrix A is orthogonal if
and only if its columns form an orthonormal basis of Rn .
Theorem 6 (5.3.4, products and inverses of orthogonal matrices).
a) The product AB of two orthogonal n × n matrices A and B is orthogonal.
b) The inverse A−1 of an orthogonal n × n matrix A is orthogonal.
Definition 7 (5.3.5). For an m × n matrix A, the transpose AT of A is the n × m matrix whose ijth entry
is the jith entry of A:
[AT ]ij = Aji .
The rows of A become the columns of AT , and the columns of A become the rows of AT .
A square matrix A is symmetric if AT = A and skew-symmetric if AT = −A.
Note 8 (5.3.6). If v and w are two (column) vectors in Rn , then
v · w = vT w.
(Here we choose to ignore the difference between a scalar a and the 1 × 1 matrix a ).
Theorem 9 (5.3.7, transpose criterion for orthogonal matrices). An n × n matrix A is orthogonal if
and only if AT A = In or, equivalently, if A has inverse A−1 = AT .
Theorem 10 (5.3.8, summary: orthogonal matrices). For an n × n matrix A, the following statements
are equivalent:
1. A is an orthogonal matrix.
2. ||Ax|| = ||x|| for all x ∈ Rn .
1
3. The columns of A form an orthonormal basis of Rn .
4. AT A = In .
5. A−1 = AT .
Theorem 11 (5.3.9, properties of the transpose).
a) If A is an n × p matrix and B a p × m matrix (so that AB is defined), then
(AB)T = B T AT .
b) If an n × n matrix A is invertible, then so is AT , and
(AT )−1 = (A−1 )T .
c) For any matrix A,
rank(A) = rank(AT ).

− w1

..
Theorem 12 (invertibility criteria involving rows). For an n × n matrix A = 
.
− wn
following are equivalent:
−


, the
−
1. A is invertible,
2. w1 , . . . , wn span Rn ,
3. w1 , . . . , wn are linearly independent,
4. w1 , . . . , wn form a basis of Rn .
Theorem 13 (column-row definition of matrix multiplication). Given matrices




− w1 −
|
|


..
A = v1 · · · vm  and B = 
,
.
|
|
− wm −
with v1 , . . . , vm , w1 , . . . , wm ∈ Rn , think of the vi as n × 1 matrices and the wi as 1 × n matrices. Then the
product of A and B can be computed as a sum of m n × n matrices:
AB = v1 w1 + · · · + vm wm =
m
X
vi wi .
i=1
Theorem 14 (5.3.10, matrix of an orthogonal projection). Let V be a subspace of Rn with orthonormal
basis u1 , . . . , um . Then the matrix of the orthogonal projection onto V is


|
|
QQT , where Q = u1 · · · um  .
|
|
2