BASIC PROPERTIES OF EIGENVALUES AND EIGEN VECTORS
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Transcript of BASIC PROPERTIES OF EIGENVALUES AND EIGEN VECTORS
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8/7/2019 BASIC PROPERTIES OF EIGENVALUES AND EIGEN VECTORS
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BBAASSIICC PPRROOPPEERRTTIISS
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EEIIGGNNEEVVAALLUUEESS
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EEIIGGEENNVVAACCTTOORRSS
PPRREEPPAARREEDD BBYY:: PPAATTEELL MMAALLAAYY DD ((0066AAEE668811))
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DYNAMIC ANALYSIS
The static and dynamic analysis of structural members and machine
components using finite element analysis results in a set of equilibrium
equations of the form
[M]{x}+[C]{x}+[K]{x} = {F} (1)
Where [M], [C] and [K] are the mass, damping and stiffness matrices,
respectively, [F] is the external force vector, and x, x and x are the acceleration,
velocity and displacement vectors respectively. At any instant of time the
equilibrium equation can be written.
{F1} + {FD} + {FE} = {F} (2)
Where {F1} are the inertia forces, {FD} are the damping forces and {FE}
are the elastic forces. All are time dependent and are equal to
{F1} = [M]{x}
{FD} = [C]{x}
{FE} = [K]{x}
Equation (1) represents a system of linear differential equations of
second order. Several solution techniques are available to solve the differential
equations with constant coefficients. Most of the solution procedures for the
general systems of differential equations are very expensive and time
consuming on the computer if the order of the matrices is large say more than
100. The coefficient matrices [M], [C] and [K] of mechanical and structural
engineering problems have special characteristics which can be effectively
utilized in simplifying the analysis. The solution techniques that are considered
here are mainly matrix methods.
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BASIC PROPERTIES OF EIGENVALUES AND EIGENVENCTORS
If [A] is any square matrix of order nxn an if {x} is a non zero vector such
that
[A]{x} = {x}
Where is some number, than {x} is said to be an eigenvector of [A] with
the corresponding eigenvalue . Premultiplying and eigenvector by the
appropriate matrix yields a constant times the eigenvector , where the
constant is the eigenvalue.
It is known that eigenvalues of a square matrix of size nxn satisfy an nth
order polynomial equation. Thus in general there will be n eigenvalues which
are not necessarily distinct and real. They are either real or result in complex
conjugates if all the matrix elements are real. Another property of eigenvalue
problem can be stated as follows. If [A] is a square matrix of size nxn, any
eigenvalue satisfies the nth degree polynomial equation: det(A - I) = 0. This
equation is also known as the characteristic equation of [A]. the proof of this
property is that we seek and non zero {x} such that
[A]{x} = {x}
[A- I}{x} = 0
This represents a system of n homogeneous equations in n unknowns
x1, x2, x3, xn. Hence if we require non zero solutions {x} then [A- I] must be
singular or det[A- I] = 0.
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det[A- I] =
det[A- I] = (a11- )(a22-).(ann-)
The equations det[A- I] = 0 gives a polynomial equation of nth degree
in . There will be n eigenvalues and n corresponding eigenvectors. The
eigenvalues are 1,2,n and the corresponding eigenvectors are {x(1)
},
{x(2)
},. {x(n)
} . that is
[A] {x(r)
} = r {xr} where 1 r n
In general equation (1) provides a procedure for the calculation of
eigenvalues and eigenvectors. The determinant of [A - I] leads to an explicit
polynomial equation. The roots of the polynomial equation give all the
eigenvalues 1,2,n. Substituting the eigen values r into equation (2) yields
n equations, [A- I] {x(r)
}= 0 the solution of which gives {x(r)
}.
a11 - a12 a1n
a22 a22- a23 a2n
an1 an2 ann-
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Example
Find eignevalues and he corresponding eigenvectors of the matrix [A].
[A] =
SOLUTION
[ A-I] =
And
det[ A-I] = (2-0{(2+) +2} 1{0(-)+2} 1{0(1)+{2+}
= -3+=0
Which is the characteristic equation.
The eigenvalues are 0, 1, -1. The corresponding eigenvectors are:
2 1 -10 -2 -2
1 1 0
2- 1 -1
0 -2- -2
1 1 -