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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
EIGENVALUES AND EIGENVECTORS.
By: Majid Khan
Department of Basic Sciences FAST University Hayat Abad
Peshawar, Pakistan
July 20, 2013
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
Outline
1 Definition
2 computing eigenvalues and eigenvectors
3 Example
4 multiplicity of an eigenvalue
5 Applications
6 History
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
DefinitionLet A be an n nmatrix. A scaler is called an
eigenvalue
of a matrix A if there is a nontrivial solution v of
Av = v
Such a v is called an eigenvector corresponding to the
eigenvalue
What does this mean geometrically?
Suppose that A is the standard matrix for a linear
transformation T : Rn Rn. Then if Ax = x, it followsthat T(x) =
x. This means that if x is an eigenvector of A,then the image of x
under the transformation T is a scalar
multiple of x and the scalar involved is the corresponding
eigenvalue . In other words, the image of x is parallel to
x.
Note that an eigenvector cannot be 0, but an eigenvalue
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
Definition cont...Suppose that 0 is an eigenvalue of A. What
does that say
about A? There must be some nontrivial vector x for which
Ax = 0x = 0
which implies that A is not invertible by Invertible Matrix
Theorem.
Invertible Matrix Theorem : The n n matrix A is invertible
if and only if 0 is not an eigenvalue of A.
If v is an eigenvector then any scalar multiple of v is alsoan
eigenvector. In fact,
If u and v are two eigenvectors for the same eigenvalue ,
then any linear combination a u+ b v is also aneigenvector with
eigenvalue .
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
Definition cont...The defining equation
Av = v
Av = Iv
Av Iv = 0
(A I)v = 0
where I is the identity matrix. Since v= 0 (by definition),
Thus v is an eigenvector of A corresponding to theeigenvalue if
and only if v and satisfy (A I)v = 0.The equation det(A I) = 0 is
called characteristicequation.
And the polynomail obained from P() = det(A I) = 0
is called characteristic polynomial of the matix A.5/ 14
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
Procedure for computing eigenvalues andeigenvectors
Compute the characteristic polynomial
Find its roots these are the eigenvalues
For each of these eigenvalues, compute the corresponding
eigenvectors. Sometimes there may be many different
eigenvectors.
DefinationThe eigenvectors corresponding to an eigenvalue form
a
subspace called the eigenspace corresponding to that
eigenvalue.
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Definition
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
For the eigenvalue = 1 the corresponding eigenvector isthe
solution of
17 1 43
9 22 1
x1x2
=
18 42
9 21
x1x2
v = 0.
so x2 = 219
x1 = 73
x1
This tells us that the eigenvectors corresponding to the
eigenvalue 1 are precisely the set of scalar multiples of
the
vector 37
In other words,the eigenspace corresponding to the
eigenvalue 1 is
span3
7
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Definition
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
For the eigenvalue = 4 the corresponding eigenvector isthe
solution of
17 4 43
9 22 4
x1x2
=
18 42
9 21
x1x2
v = 0.
This tells us that the eigenvectors corresponding to the
eigenvalue 4 are precisely the set of scalar multiples of
the
vector
2
1In other words,the eigenspace corresponding to the
eigenvalue 4 is
span
2
1
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Definition
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
Mona Lisa picture
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Definition
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Definition
computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
Contributions of Mathematicians
Euler
Lagrange
Cauchy
Fourier, Hermite, Liouville,Laplcae etc
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Definition
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computing eigenvalues and eigenvectors
Example
multiplicity of an eigenvalue
Applications
History
Questions
Questions?
Thank you
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Appendix For Further Reading
References I
[allowframebreakes]
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