MTH 101 Term Paper by Mridul

8/7/2019 MTH 101 Term Paper by Mridul

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TOPIC:Prove that the product of the two unitary matrices (of the same size)and the inverse of a unitary matrix are unitary. Give examples. Powers of

unitary matrices occurring in applications may sometimes be familiar real

matrices.

SUMITTED TO:- SUBMITTED

BY:-

Amarpreet SinghMridul Mudai

B.Tech(ME)

RollNo-RB4004B64

2010 Mridul

TERM PAPER

OF

ENGINEERING

MATHEMATICS


2/15

Regd. No-11010983

Acknowledgement:This project is a welcome and challenging experience for us as it took a great

deal of hard work and dedication for its successful completion. Its my pleasure

to take this opportunity to thank all those who helped me directly or indirectly

in preparation of this report.

I would also like to very sincerely thank to my project guide Lecturer Mr.

Amarpreet Singh who supported me technically as well as morally in every

stage of the project and also gave his very valuable suggestions to me which

helped me to complete my project. Without his guidance and support, this

project would not have seen light of the day.

It gives me immense in expressing a deep sense of gratitude and sincere

thanks to Lovely Professional University.

Last but not the least I thank my family for their boost and support in every

sphere. Their vital push infused a sense of insurgency in me.

Mridul Mudai

2010 Mridul


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Table Of Contents:

Introduction

Properties Of Unitary Matrices

Proof Showing that the product of the two unitary matrices (of the samesize) and the inverse of a unitary matrix are unitary

Examples

Powers of unitary matrices occurring in applications may sometimes befamiliar real matrices

External Links

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1)INTRODUCTION:

What is a unitary matrix?

A square matrix U is a unitary matrix if

(1)

where denotes the conjugate transpose and is the matrix inverse. For

example,

(2)

is a unitary matrix.

Unitary matrices leave the length of a complex vectorunchanged.

Forreal matrices, unitary is the same as orthogonal. In fact, there are some

similarities between orthogonal matrices and unitary matrices. The rows of a

unitary matrix are a unitary basis. That is, each row has length one, and their

Hermitian inner product is zero. Similarly, the columns are also a unitary basis.

In fact, given any unitary basis, the matrix whose rows are that basis is a unitary

matrix. It is automatically the case that the columns are another unitary basis.

A matrix m an be tested to see if it is unitary using the Mathematica function:

2010 Mridul
http://mathworld.wolfram.com/SquareMatrix.htmlhttp://mathworld.wolfram.com/ConjugateTranspose.htmlhttp://mathworld.wolfram.com/MatrixInverse.htmlhttp://mathworld.wolfram.com/ComplexVector.htmlhttp://mathworld.wolfram.com/RealMatrix.htmlhttp://mathworld.wolfram.com/OrthogonalMatrix.htmlhttp://mathworld.wolfram.com/OrthogonalMatrix.htmlhttp://mathworld.wolfram.com/UnitaryBasis.htmlhttp://mathworld.wolfram.com/HermitianInnerProduct.htmlhttp://www.wolfram.com/products/mathematica/http://mathworld.wolfram.com/SquareMatrix.htmlhttp://mathworld.wolfram.com/ConjugateTranspose.htmlhttp://mathworld.wolfram.com/MatrixInverse.htmlhttp://mathworld.wolfram.com/ComplexVector.htmlhttp://mathworld.wolfram.com/RealMatrix.htmlhttp://mathworld.wolfram.com/OrthogonalMatrix.htmlhttp://mathworld.wolfram.com/OrthogonalMatrix.htmlhttp://mathworld.wolfram.com/UnitaryBasis.htmlhttp://mathworld.wolfram.com/HermitianInnerProduct.htmlhttp://www.wolfram.com/products/mathematica/


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UnitaryQ[m_List?MatrixQ] :=

(Conjugate@Transpose @ m . m ==

IdentityMatrix @ Length @ m)

The definition of a unitary matrix guarantees that

(3)

where is the identity matrix.

In particular, a unitary matrix is always invertible, and . Note that

transpose is a much simpler computation than inverse. A similarity

transformation of a Hermitian matrix with a unitary matrix gives

(4)

(5)

(6)

(7)

(8)

Unitary matrices are normal matrices. If is a unitary matrix, then the

permanent

(9)

The unitary matrices are precisely those matrices which preserve the Hermitian

inner product

(10)

Also, the norm of the determinant of is . Unlike the orthogonal

matrices, the unitary matrices are connected. If then is a special unitary

matrix.

The product of two unitary matrices is another unitary matrix. The inverse of a

unitary matrix is another unitary matrix, and identity matrices are unitary.

Hence the set of unitary matrices form a group, called the unitary group.

If is an n by n matrix then the following are all equivalent conditions:

2010 Mridul
http://mathworld.wolfram.com/IdentityMatrix.htmlhttp://mathworld.wolfram.com/Transpose.htmlhttp://mathworld.wolfram.com/SimilarityTransformation.htmlhttp://mathworld.wolfram.com/SimilarityTransformation.htmlhttp://mathworld.wolfram.com/HermitianMatrix.htmlhttp://mathworld.wolfram.com/NormalMatrix.htmlhttp://mathworld.wolfram.com/Permanent.htmlhttp://mathworld.wolfram.com/HermitianInnerProduct.htmlhttp://mathworld.wolfram.com/HermitianInnerProduct.htmlhttp://mathworld.wolfram.com/OrthogonalMatrix.htmlhttp://mathworld.wolfram.com/OrthogonalMatrix.htmlhttp://mathworld.wolfram.com/ConnectedSpace.htmlhttp://mathworld.wolfram.com/SpecialUnitaryMatrix.htmlhttp://mathworld.wolfram.com/SpecialUnitaryMatrix.htmlhttp://mathworld.wolfram.com/IdentityMatrix.htmlhttp://mathworld.wolfram.com/Group.htmlhttp://mathworld.wolfram.com/UnitaryGroup.htmlhttp://mathworld.wolfram.com/IdentityMatrix.htmlhttp://mathworld.wolfram.com/Transpose.htmlhttp://mathworld.wolfram.com/SimilarityTransformation.htmlhttp://mathworld.wolfram.com/SimilarityTransformation.htmlhttp://mathworld.wolfram.com/HermitianMatrix.htmlhttp://mathworld.wolfram.com/NormalMatrix.htmlhttp://mathworld.wolfram.com/Permanent.htmlhttp://mathworld.wolfram.com/HermitianInnerProduct.htmlhttp://mathworld.wolfram.com/HermitianInnerProduct.htmlhttp://mathworld.wolfram.com/OrthogonalMatrix.htmlhttp://mathworld.wolfram.com/OrthogonalMatrix.htmlhttp://mathworld.wolfram.com/ConnectedSpace.htmlhttp://mathworld.wolfram.com/SpecialUnitaryMatrix.htmlhttp://mathworld.wolfram.com/SpecialUnitaryMatrix.htmlhttp://mathworld.wolfram.com/IdentityMatrix.htmlhttp://mathworld.wolfram.com/Group.htmlhttp://mathworld.wolfram.com/UnitaryGroup.html


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1. is unitary2. is unitary3. the columns of form an orthonormal basis of with respect to this

inner product

4. the rows of form an orthonormal basis of with respect to this innerproduct

5. is an isometry with respect to the norm from this inner product6. is a normal matrix with eigenvalues lying on the unit circle.

2)PROPERTIES OF UNITARY

MATRICES: All unitary matrices are normal, and the spectral theorem therefore

applies to them. Thus every unitary matrix Uhas a decomposition of the

form

where V is unitary, and is diagonal and unitary. That is, a unitary matrix is

diagonalizable by a unitary matrix.

For any unitary matrix U, the following hold:

is invertible, with . is also unitary.

preserves length ("isometry"): .

has complex eigenvalues of modulus 1.[1]

For any n, the set of all n by n unitary matrices with matrix multiplication

forms a group, called U(n).

Any unit-norm matrix is the average of two unitary matrices. As a consequence,

every matrix M is a linear combination of two unitary matrices.

.The determinant of a unitary matrix is a complex number of modulus one.

2010 Mridul
http://en.wikipedia.org/wiki/Orthonormal_basishttp://en.wikipedia.org/wiki/Isometryhttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Eigenvalueshttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Spectral_theoremhttp://en.wikipedia.org/wiki/Diagonalizable_matrixhttp://en.wikipedia.org/wiki/Group_(mathematics)http://en.wikipedia.org/wiki/Unitary_grouphttp://en.wikipedia.org/wiki/Orthonormal_basishttp://en.wikipedia.org/wiki/Isometryhttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Eigenvalueshttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Spectral_theoremhttp://en.wikipedia.org/wiki/Diagonalizable_matrixhttp://en.wikipedia.org/wiki/Group_(mathematics)http://en.wikipedia.org/wiki/Unitary_group


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3)PROOF SHOWING THAT THE

PRODUCT OF TWO UNITARY

MATRICES(OF THE SAME SIZE)AN D

THE INVERSE OF A UNITARY MATRIXARE UNITARY MATRIX WITH

EXAMPLES

Proof By Different Ways:

1st method:

Let A, B and C be unitary matrices

we want to show [AB(C-1)]*(ABC(-1)) = I

[AB(C-1)]*(ABC(-1)) = (c-1)*B*A*ABC-1

but A*A = I, B*B = I so we are left with

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(C-1)*(C-1)

but C is a unitary matrix --> (c-1)=C*

so (C-1)*(C-1) = (C*)*(C*) = CC* = I

so [AB(C-1)]*(ABC(-1))=I

The exact same thing works for showing (ABC(-1)) [AB(C-1)]* = I.

2nd method:

If A and B are unitary, then

A^(-1) = A^* and B^(-1) = B^*

where "^*" is used to indicate the conjugate transpose.

For any matrices U and V, (UV)^* = V^* U^*. Considering the unitary matrices

A and B above, note that

(AB)^* = B^*A^*.

So show that (AB)^* is in fact (AB)^(-1), observe that

(AB)^*(AB) = B^*A^*(AB) = B^*(A^*A)B = B^*IB = B^*B = I

hence (AB)^* = (AB)^(-1).

The second result is sort of trivial. For any matrix U

(U^*)^* = U.

For unitary matrix A, A^(-1) = A^*, that is, A^* is the inverse of A. As

(A^*)^* = A

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and as A is the inverse of A^*, we see that A^(-1) (which is A^*) is also

unitary.

3rd Method:

Let X be the conjugate transpose of the matrix X.

Then X is unitary if and only if X X = X X = I

If A and B are unitary matrices, then (AB) (AB) = B A A B = B I B = Iand similarly, (AB) (AB) = I. So AB is also unitary.

If A is a unitary matrix, then A A = A A = I. So clearly A is the inverse ofA. It follows that the inverse of A is unitary.

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4)EXAMPLES:

Example:- (i) (1+i) (-1+i)

Let A= (1+I ) ( 1-I )

= (1-i) (-1-i)

(1-i) (1+i)

( ) /= (1-i) (1-i)

(-1-i) (1+i)

|A|= 1/2 (1+i)*1/2 (1-i)-1/2 (1+i)*1/2 (-1+i)

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= 1/4{(1+i)*(1-i)-(1+i)*(i-1)}

= 1/4{1- i2 -( i2-1)}

= {1-i2-i2+1}

= {2-2i2}

= {2+2} = 1

(1-i) (1-i)

A-1 =

(-1-i) (1+i)

hence (A-1

) = A-1

Hence A is unitary matrix.

Now A-1 = (1-i) (1-i) =B (Assume)

(-1-i) (1+i)

B = (1+i) (1+i)

(-1+i) (1-i)

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(B) = (1+i) (-1+i)

(1+i) (1-i)

|B| = 1

B-1 = (1+i) (-1+i)

(1+i) (1-i)

Hence B-1 = (B)-1

Hence B i.e A-1 is also a unitary matrix

Hence A-1 is also a unitary matrix.

Proving of second property :-

Let A=(a+ib)nn & B=(c+id)nn be the unitary matrices,then

(A) = A-1..............(1)

(B)-1 = B-1.(2)

Let AB=C where C is a matrix formed by multiplication of A&B.

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Now (C)/= (AB)/

= (A B )/

= (B)/ (A)/.(3)

As (AB)/= B/A/

A & B are unitary matrices then

(C)/ = B-1 A-1

= (AB)-1 as (AB)-1 = B-1 A-1

(C)/ = C-1

Hence C is a unitary matrix.

Example:- (ii)

Let A = (1-i) (-1-i) B =1/6 -4 -2-4i

(1-i) (1+i) 2-4i -4

=1/2 1-i -1-i =1/3 -2 -1-2i

1-i 1+i 1-2i -2

AB = 1/6 2i-2-1-i+2i-2 i-1+2+2i

2i-2+1-2i+i+2 i-1-2-2i

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AB =1/6 3i-5 3i+1

i+1 -i-3

(AB)/ =1/6 3i-5 i+1

3i+1 -i-3

(AB)/ =1/6 -3i-5 -i+1

-3i+1 i-3

(AB)-1 =1/6 -3i-5 -i+1

-3i+1 i-3

Hence (AB)/ = (AB)-1 ,hence AB is also a unitary matrix

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6)External Links:.

Ivanova, O.A. (2001), "Unitary matrix", in Hazewinkel, Michiel

http://eom.springer.de/U/u095540.htm

http://www.google.com

http://mathworld.wolfram.com/UnitaryMatrix.html

Higher Engineering mathematics-Grewal

2010 Mridul
http://eom.springer.de/U/u095540.htmhttp://eom.springer.de/U/u095540.htmhttp://mathworld.wolfram.com/UnitaryMatrix.htmlhttp://eom.springer.de/U/u095540.htmhttp://eom.springer.de/U/u095540.htmhttp://mathworld.wolfram.com/UnitaryMatrix.html

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