1) Prepared By : Pratik Sharma, Smit Shah, Shivani Sharma, Rehan Shaikh Smarth Shah ENROLLMENT NO. : 140410109098 Branch :Electrical Engineering Guided by: H PATEL
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALIZATION.
Jan 15, 2017
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1) Prepared By : Pratik Sharma, Smit Shah, Shivani Sharma, Rehan Shaikh Smarth ShahENROLLMENT NO. : 140410109098 140410109096 140410109099 140410109097 140410109095
Branch :Electrical Engineering
Guided by:H PATEL
ORTHOGONAL VECTOR.ORTHONORMAL VECTOR.GRAM SCHMIDT PROCESS.ORTHOGONALLY DIAGONALIZATION.
CONTENTS
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.
ORTHOGONAL VECTORS
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.
EXAMPLE:
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.
ORTHONORMAL VECTOR
jijiVS
ji
n
01
,
,,, 21
vv
vvv
E.g.1:- Check whether this is orthonormal or not ?
~
so,We can say that given vector is orthogonal.
EXAMPLE:
31,
32,
32,
322,
62,
62,0,
21,
21
321
S
vvv
1||||
1||||
10||||
91
94
94
333
98
362
362
222
21
21
111
vvv
vvv
vvv
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:
GRAM SCHMIDT PROCESS
Gram-Schmidt process:- is a basis for an inner product
space V },,,{ 21 nB uuu
11Let uv
},,,{' 21 nB vvv
},,,{''2
2
n
nBvv
vv
vv
1
1
1
1 〉〈〉〈proj
1
n
ii
ii
innnnn n
vv,vv,vuuuv W
2
22
231
11
133333 〉〈
〉〈〉〈〉〈proj
2v
v,vv,uv
v,vv,uuuuv W
111
122222 〉〈
〉〈proj1
vv,vv,uuuuv W
is an orthogonal basis.
is an orthonormal basis.
)0,1,1(11 uv
)2,0,0()0,21,
21(
2/12/1)0,1,1(
21)2,1,0(
222
231
11
1333
vvvvuv
vvvuuv
)}2,1,0(,)0,2,1(,)0,1,1{(321
Buuu
)0,21,
21()0,1,1(
23)0,2,1(1
11
1222
vvvvuuv
Sol:
EXAMPLE
Ex1 : (Applying the Gram-Schmidt orthonormalization process)
Apply the Gram-Schmidt process to the following basis.
}2) 0, (0, 0), , 21 ,
21( 0), 1, (1,{},,{' 321
vvvB
}1) 0, (0, 0), , 2
1 ,21( 0), ,
21 ,
21({},,{''
3
3
2
2
vv
vv
vv
1
1B
Orthogonal basis
Orthonormal basis
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.
ORTHOGONAL DIAGONALIZATION
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]
How to orthogonally diagonalize a matrix?
13/80
14/80
15/80
16/80
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