S T
7
# $ W≠4X
xÕ y = 1 2 3 4 WU 3 V = 7 ≠ 8 + 9 ≠ 8 = 0.
X
≠2
Example:
5 6 Let 5 6
1 2
x= and y = .
2 ≠1
Now inner product between the vectors x and y is
xÕ y = 0
This means that vectors form a right angle.
Right angle
Note: Vectors u and v are called orthonormal, if vectors are unit vectors (||u|| = ||v|| = 1) and vectors are
also orthogonal (i.e. inner product between the vectors is zero uÕ v = 0).
Example: Consider vectors
5 6 5 6
1 2
x= and y =
2 ≠1
Next we will form normalized vectors u and v.
Normalized vector of x is a vector in the same direction but with norm 1. Normalized vector is denoted as
x
u=
||x||
S 1 T
5 6 Ô
x 1 1
= U 25 V
W X
u= =Ô
||x|| 5 2 Ô
5
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and normalized vector of y is called as v
S 2 T
5 6
Ô
y 1 2
= U ≠15 V
W X
v= =Ô
||y|| 5 ≠1 Ô
5
Now vectors u and v are also orthogonal since uÕ v = 0. This is obvious since we noriced in a previous exmple
that vectors x and y are orthogonal and that normalization does not change direction. Since vectors u and v
are unit vectors and orthogonal they are orthonormal.
# R example of orthonormal vector
# Define vector x
x <- c(1,2)
# compute the length of the vector x
sqrt(x%*%x)
## [,1]
## [1,] 2.236068
# Create normalized vector of x that is called as u
u <- x/sqrt(x%*%x)
## Warning in x/sqrt(x %*% x): Recycling array of length 1 in vector-array arithmetic is deprecated.
## Use c() or as.vector() instead.
u
## [1] 0.4472136 0.8944272
# Define vector y
y <- c(2,-1)
# Create normalized vector of y that is called as v
v <- y/sqrt(y%*%y)
## Warning in y/sqrt(y %*% y): Recycling array of length 1 in vector-array arithmetic is deprecated.
## Use c() or as.vector() instead.
v
## [1] 0.8944272 -0.4472136
# Vectors u and v are orthonormal if inner product between the vectors is zero.
u%*%v
## [,1]
## [1,] 0
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Note: The set of vectors {u1 , u2 , ..., up } is called orthogonal set of vectors if all possible pairs of vectors are
orthogonal. In other words
uiÕ uj = 0 ’ i ”= j.
Example: In this example we investigate if the set of vectors is orthogonal.
Consider the set of three vectors {u1 , u2 , u3 }, where
# $
u1T = 3 1 1 ,
# $
u2T = ≠1 2 1
and
# $
u3T = ≠1/2 ≠2 7/2
We investigate if the vector pairs are orthogonal.
Are vectors u1 and u2 orthogonal?
u1T u2 = 3 · (≠1) + 1 · 2 + 1 · 1 = 0
Yes they are, since inner product between the vectors is zero.
Pair u1 and u3 is also orthogonal since
u1T u3 = 3 · (≠1/2) + 1 · (≠2) + 1 · 7/2 = 0.
and the vectors u2 and u3 are also orthogonal since
u2T u3 = ≠1 · (≠1/2) + 2 · (≠2) + 1 · 7/2 = 0.
We confirmed that all possible pairs of vectors are orthogonal so the set of vectors {u1 , u2 , u3 } is orthogonal.
Note: The set of vectors is orthonormal if the set of vectors is orthogonal and all vectors are unit vectors.
To be more precise the set of vectors {u1 , u2 , ..., up } is orthonormal if
;
1, if i = j
uiT uj =
0, if i =
” j
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