8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

1/13

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

2/13

ACKNOWLEDGEMENT

It acknowledges all the contributors involved in the preparation of this project. Including me,

there is a hand of my teachers, some books and internet. I express most gratitude to my

subject teacher, who guided me in the right direction. The guidelines provided by her helped

me a lot in completing the assignment.

The books and websites I consulted helped me to describe each and every point mentioned in

this project. Help of original creativity and illustration had taken and I have explained each

and every aspect of the project precisely.

At last it acknowledges all the members who are involved in the preparation of this project.

Thanks

ANKIT VERMA

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

3/13

ABSTRACT

In mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field oflinear

algebra. The prefix eigen- is adopted from the German word "eigen" for "innate", "idiosyncratic",

"own".[1]Linear algebra studies linear transformations, which are represented by matrices acting

onvectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They are computed

by a method described below, give important information about the matrix, and can be used in matrix

factorization. They have applications in areas of applied mathematics as diverse

aseconomics and quantum mechanics.

In general, a matrix acts on a vectorby changing both its magnitude and its direction. However, a

matrix may act on certain vectors by changing only their magnitude, and leaving their direction

unchanged (or possibly reversing it). These vectors are the eigenvectors of the matrix. A matrix acts

on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is

unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with that

eigenvector. An eigenspace is the set of all eigenvectors that have the same eigenvalue, together with

the zero vector.

http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wiktionary.org/wiki/de:eigenhttp://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace#cite_note-0http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace#cite_note-0http://en.wikipedia.org/wiki/Linear_transformationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Coordinate_vectorhttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_factorizationhttp://en.wikipedia.org/wiki/Matrix_factorizationhttp://en.wikipedia.org/wiki/Mathematical_economicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Vector_(mathematics)http://en.wikipedia.org/wiki/Magnitude_(vector)http://en.wikipedia.org/wiki/Direction_(geometry)http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wiktionary.org/wiki/de:eigenhttp://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace#cite_note-0http://en.wikipedia.org/wiki/Linear_transformationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Coordinate_vectorhttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_factorizationhttp://en.wikipedia.org/wiki/Matrix_factorizationhttp://en.wikipedia.org/wiki/Mathematical_economicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Vector_(mathematics)http://en.wikipedia.org/wiki/Magnitude_(vector)http://en.wikipedia.org/wiki/Direction_(geometry)

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

4/13

TABLE OF CONTENT

1. EIGEN VALUE

2. COMPUTATION OF EIGEN VALUES AND THE CHARACTERSTICS

EQUATION

3. HERMITTIAN MATRIX

4. PROPERTIES OF HERMITTTIAN MATRIX

5. PROOF:EIGEN VALUE OF HERMITTIAN MATRIX IS REAL

6. EXAMPLE OF ABOVE

7. DET (A-3I)

8. BIBLOGRAPHY

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

5/13

Eigenvalues

If the action of a matrix on a (nonzero) vector changes its magnitude but not its direction,

then the vector is called an eigenvector of that matrix. A vector which is "flipped" to point in

the opposite direction is also considered an eigenvector. Each eigenvector is, in effect,multiplied by a scalar, called the eigenvalue corresponding to that eigenvector. The

eigenspace corresponding to one eigenvalue of a given matrix is the set of all eigenvectors of

the matrix with that eigenvalue.

Definition

Given a linear transformationA, a non-zero vector x is defined to be an

eigenvectorof the transformation if it satisfies the eigenvalue equation

for some scalar. In this situation, the scalar is called an eigenvalue ofA

corresponding to the eigenvector x.

The key equation in this definition is the eigenvalue equation,Ax = x. That is to say that the

vector x has the property that its direction is not changed by the transformationA, but that it

is only scaled by a factor of . Most vectors x will not satisfy such an equation: a typical

vector x changes direction when acted on byA, so thatAx is not a multiple of x. This means

that only certain special vectors x are eigenvectors, and only certain special scalars are

eigenvalues. Of course, ifA is a multiple of the identity matrix, then no vector changes

direction, and all non-zero vectors are eigenvectors.

http://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Identity_matrix

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

6/13

Computation of eigenvalues, and the characteristic

equation

When a transformation is represented by a square matrixA, the eigenvalue equation can be

expressed as

This can be rearranged to

If there exists an inverse

then both sides can be left multiplied by the inverse to obtain the trivial solution: x = 0. Thus

we require there to be no inverse by assuming from linear algebra that the determinant equalszero:

det(A I) = 0.

The determinant requirement is called the characteristic equation (less often, secular

equation) ofA, and the left-hand side is called thecharacteristic polynomial. When

expanded, this gives apolynomial equation for . The eigenvector x or its components are not

present in the characteristic equation.

The matrix

defines a linear transformation of the real plane. The eigenvalues of this transformation are

given by the characteristic equation

http://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Characteristic_equationhttp://en.wikipedia.org/wiki/Secular_equationhttp://en.wikipedia.org/wiki/Secular_equationhttp://en.wikipedia.org/wiki/Characteristic_polynomialhttp://en.wikipedia.org/wiki/Characteristic_polynomialhttp://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Characteristic_equationhttp://en.wikipedia.org/wiki/Secular_equationhttp://en.wikipedia.org/wiki/Secular_equationhttp://en.wikipedia.org/wiki/Characteristic_polynomialhttp://en.wikipedia.org/wiki/Polynomial

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

7/13

The roots of this equation (i.e. the values of for which the equation holds) are = 1 and =

3. Having found the eigenvalues, it is possible to find the eigenvectors. Considering first the

eigenvalue = 3, we have

After matrix-multiplication

This matrix equation represents a system of two linear equations 2x +y = 3x andx + 2y = 3y.

Both the equations reduce to the single linear equationx =y. To find an eigenvector, we are

free to choose any value for x (except 0), so by picking x=1 and setting y=x, we find aneigenvector with eigenvalue 3 to be

We can confirm this is an eigenvector with eigenvalue 3 by checking the action of the matrix

on this vector:

Any scalar multiple of this eigenvector will also be an eigenvector with eigenvalue 3.

For the eigenvalue = 1, a similar process leads to the equation x = y, and hence an

eigenvector with eigenvalue 1 is given by

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

8/13

Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square

matrix with complex entries that is equal to its own conjugate transpose that is, the element

in the i-th row andj-th column is equal to the complex conjugate of the element in thej-th

row and i-th column, for all indices i andj:

If the conjugate transpose of a matrixA is denoted by , then the Hermitian property

can be written concisely as

Hermitian matrices can be understood as the complex extension of real symmetric

matrices.

For example,

is a Hermitian matrix.

http://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Symmetric_matrix

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

9/13

Properties of hermittian matrix

The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are

necessarily real. A matrix that has only real entries is Hermitian if and only ifit isa symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and

symmetric matrix is simply a special case of a Hermitian matrix.

Every Hermitian matrix is normal, and the finite-dimensional spectral theorem applies. It says

that any Hermitian matrix can be diagonalizedby a unitary matrix, and that the resulting

diagonal matrix has only real entries. This means that all eigenvalues of a Hermitian matrix

are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal. It is possible to

find an orthonormal basis of Cnconsisting only of eigenvectors.

The sum of any two Hermitian matrices is Hermitian, and the inverse of an invertible

Hermitian matrix is Hermitian as well. However, theproduct of two Hermitian

matricesA andB will only be Hermitian if they commute,

The eigenvectors of a Hermitian matrix are orthogonal, i.e.,

its eigendecomposition is where Since right- and left- inverse are

the same, we also have

and therefore

,

where i are the eigenvalues and ui the eigenvectors.

Additional properties of Hermitian matrices include:

The sum of a square matrix and its conjugate transpose is

Hermitian.

http://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Spectral_theoremhttp://en.wikipedia.org/wiki/Diagonalizable_matrixhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Eigenvectorshttp://en.wikipedia.org/wiki/Eigenvectorhttp://en.wikipedia.org/wiki/Orthonormal_basishttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrixhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Spectral_theoremhttp://en.wikipedia.org/wiki/Diagonalizable_matrixhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Eigenvectorshttp://en.wikipedia.org/wiki/Eigenvectorhttp://en.wikipedia.org/wiki/Orthonormal_basishttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

10/13

The difference of a square matrix and its conjugate transpose

is skew-Hermitian (also called antihermitian).

An arbitrary square matrix Ccan be written as the sum of a Hermitian

matrixA and a skew-Hermitian matrixB:

The determinant of a Hermitian matrix is real:

Proof:

Therefore if

Theorem:Eigenvalues of a Hermitian matrix are real

Proof:

Suppose is a (non-zero) eigenvectorof with eigenvalue ,

.

,

,

,

.

Thus either , contrary to assumption, or .

Example

A is Hermitian matrix

A = | 1 1+i| This is Hermitian.

|1-i 2 |

CHARACTERSTIC EQUATION OF EIGEN VALUE

|A-KI|=0

\Its characteristic equation is |1-k 1+i| = 0

http://en.wikipedia.org/wiki/Skew-Hermitian_matrixhttp://www.mathematics.thetangentbundle.net/w/index.php?title=non-zero&action=edithttp://www.mathematics.thetangentbundle.net/wiki/Linear_algebra/eigenvectorhttp://www.mathematics.thetangentbundle.net/w/index.php?title=Linear_algebra/eigenvalue&action=edithttp://en.wikipedia.org/wiki/Skew-Hermitian_matrixhttp://www.mathematics.thetangentbundle.net/w/index.php?title=non-zero&action=edithttp://www.mathematics.thetangentbundle.net/wiki/Linear_algebra/eigenvectorhttp://www.mathematics.thetangentbundle.net/w/index.php?title=Linear_algebra/eigenvalue&action=edit

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

11/13

|1-i 2-k|

which gives k1 = 0 and k2 = 3 (both real)

Det (H-3I) can not be zero where H is the hermittian matrix

and I is the unit matrix

H = | a b+i|

|b-i c |

|H-3I|= | a-3 b+i|

|b-i c-3 |

=(a-3)(c-3)-(b2-i2)

=(a-3)(c-3)-(b2+1)

Where a, b, c are real and positive

We will obtain the real value not zero

For example

H = | 3 2+i||2-i 1 |

|H-3I|= | 3-3 2+i|

|2-i 1-3 |

= -3

Hence proved

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

12/13

BIBLOGRAPHY

1. www.mathfax.com

2. www.mathforum.com

3. HIGHER ENGINEERING MATHEMATICS -B.V.RAMANA

4. HIGHER ENGINEERING MATHEMATICS -G.S.GREWAL

http://www.mathfax.com/http://www.mathfax.com/http://www.mathfax.com/

8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-

13/13