Page 298 - Matrix Analysis & Applied Linear Algebra

294 Chapter 5 Norms, Inner Products, and Orthogonality
5.4 ORTHOGONAL VECTORS

3
Two vectors in are orthogonal (perpendicular) if the angle between them is
a right angle (90 ). But the visual concept of a right angle is not at our disposal in
◦
higher dimensions, so we must dig a little deeper. The essence of perpendicularity
2
3
in and is embodied in the classical Pythagorean theorem,
v || u - v ||

u
|| v ||
|| u ||

2 2 2
which says that u and v are orthogonal if and only if u + v = u − v .
2
T
3
T
T
But 39 u = u u for all u ∈ , and u v = v u, so we can rewrite the
Pythagorean statement as
2 2 2 T T T
0 = u + v − u − v = u u + v v − (u − v) (u − v)

T
T
T
T
T
T
T
= u u + v v − u u − u v − v u + v v =2u v.
T
3
Therefore, u and v are orthogonal vectors in if and only if u v =0. The
natural extension of this provides us with a definition in more general spaces.
Orthogonality
In an inner-product space V, two vectors x, y ∈V are said to be
orthogonal (to each other) whenever x y =0, and this is denoted
by writing x ⊥ y.
n T
• For with the standard inner product, x ⊥ y ⇐⇒ x y =0.
n
∗
• For C with the standard inner product, x ⊥ y ⇐⇒ x y =0.
Example 5.4.1
   
1 4
−2 1 T
x =   is orthogonal to y =   because x y =0.
3 −2
−1 −4
39 2
Throughout this section, only norms generated by an underlying inner product =
are used, so distinguishing subscripts on the norm notation can be omitted.