prove that eigen value of hermittian matrix are alays real.give example.use the above result to show...
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8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-
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8/6/2019 prove that eigen value of hermittian matrix are alays real.give example.use the above result to show that det (H-
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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
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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) -
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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
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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 -
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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 -
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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
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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-
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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 -
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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 -
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|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
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BIBLOGRAPHY
1. www.mathfax.com
2. www.mathforum.com
3. HIGHER ENGINEERING MATHEMATICS -B.V.RAMANA
4. HIGHER ENGINEERING MATHEMATICS -G.S.GREWAL
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