The document discusses concepts related to orthogonal and orthonormal vectors, including definitions and examples such as the Gram-Schmidt process for orthogonalization. It outlines the procedure for converting linearly independent vectors into an orthonormal set and explains orthogonal diagonalization of matrices. Key steps involve finding eigenvalues, eigenvectors, and applying the Gram-Schmidt process to achieve orthonormal vectors.

1. 1) Prepared By : Pratik Sharma,
Smit Shah,
Shivani Sharma,
Rehan Shaikh
Smarth Shah
ENROLLMENT NO. : 140410109098
140410109096
140410109099
140410109097
140410109095
Branch :Electrical Engineering
Guided by:
H PATEL

2.  ORTHOGONAL VECTOR.
 ORTHONORMAL VECTOR.
 GRAM SCHMIDT PROCESS.
 ORTHOGONALLY
DIAGONALIZATION.

3.  Definition. We say that 2 vectors are
orthogonal if they are perpendicular to each
other. i.e. the dot product of the two vectors
is zero.
 Definition. We say that a set of vectors {v1,
v2, ..., vn} are mutually orthogonal if every
pair of vectors is orthogonal. i.e. vi.vj = 0, for
all i not equal j.

4. E.g.:- Check whether v1=(1,-1,0) v2=(1,1,0)
v3=(0,0,1) is orthogonal vector or not ?
~ v1.v2 =1+(-1)+0 =0
v2.v3 = 0+0+0 = 0
v3.v1 = 0+0+0 =0
so,We can say that given vector is
orthogonal.

5.  Definition. If v1,v2,……Vn whose
||v1||=||v2||=……||Vn||=1 then v1,v2,……Vn
is a orthonormal vector.
• Definition. An orthogonal set in which
each vector is a unit vector is called
orthonormal.
 







ji
ji
VS
ji
n
0
1
,
,,, 21
vv
vvv

6.  E.g.1:- Check whether this is orthonormal or not
?
 ~
 so,We can say that given vector is orthogonal.




























3
1
,
3
2
,
3
2
,
3
22
,
6
2
,
6
2
,0,
2
1
,
2
1
321
S
vvv
1||||
1||||
10||||
9
1
9
4
9
4
333
9
8
36
2
36
2
222
2
1
2
1
111



vvv
vvv
vvv

7.  If u1,u2,u3 is not orthogonal. Then v1.v2 also
not equal to 0. then we use Gram Schmidt
Produces.
 Given a set of linearly independent vectors, it
is often useful to convert them into an
orthonormal set of vectors. We first define
the projection operator.
 Definition. Let u and v be two vectors. The
projection of the vector v on u is defined as
follows:

8.  Gram-Schmidt process:-
is a basis for an inner product space
V
},,,{ 21 nB uuu 
11Let uv 
},,,{' 21 nB vvv 
},,,{''
2
2
n
n
B
v
v
v
v
v
v
1
1




 
1
1 〉〈
〉〈
proj 1
n
i
i
ii
in
nnnn n
v
v,v
v,v
uuuv W

2
22
23
1
11
13
3333
〉〈
〉〈
〉〈
〉〈
proj 2
v
v,v
v,u
v
v,v
v,u
uuuv W 
1
11
12
2222
〉〈
〉〈
proj 1
v
v,v
v,u
uuuv W 
is an orthogonal basis.
is an orthonormal basis.

9. )0,1,1(11  uv
)2,0,0()0,
2
1
,
2
1
(
2/1
2/1
)0,1,1(
2
1
)2,1,0(
2
22
23
1
11
13
33






 v
vv
vu
v
vv
vu
uv
)}2,1,0(,)0,2,1(,)0,1,1{(
321
B
uuu
)0,
2
1
,
2
1
()0,1,1(
2
3
)0,2,1(1
11
12
22 


 v
vv
vu
uv
Sol:
 EXAMPLE
 Ex1 : (Applying the Gram-Schmidt orthonormalization
process)
Apply the Gram-Schmidt process to the following basis.

10. }2)0,(0,0),,
2
1
,
2
1
(0),1,(1,{},,{' 321

 vvvB
}1)0,(0,0),,
2
1
,
2
1
(0),,
2
1
,
2
1
({},,{''
3
3
2
2 

v
v
v
v
v
v
1
1
B
Orthogonal basis
Orthonormal basis

11.  Definition. A square matrix A is orthogonally
diagonalizable if there exists an orthogonal
matrix Q such that Q^T A Q = D is a diagonal
matrix.

12.  STEP 1. find out eigen values.
 STEP 2. Find out eigen vectors.
 STEP 3. say eigen vectors as a u1,u2,u3…..
 STEP 4. convert in a ortho normal vector
q1,q2,q3….using gram schmidt process.
 STEP 5. find matrix p=[ q1,q2,q3]
 STEP 6.find p^-1 A P= P^T A P=[ λ1 0 0 ]
 [ 0 λ2 0 ]
 [ 0 0 λ3]

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