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?

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