SKMM 3023 Applied Numerical Methods
Solution of Matrix Eigenproblems
ibn ‘Abdullah
Faculty of Mechanical Engineering
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 1 / 31
Outline
1 Introduction
2 Engineering Applications
3 Definitions and Basic Facts
Eigenvalue & Eigenvector
Characteristic Equation
Standard & General Eigenvalue Problems
4 Power Method
5 Faddeev-Leverrier Method
6 Matlab eig Function
7 Bibbliographyibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 2 / 31
Introduction
In many engineering problems, it may became necessary to formulate clearly thedifference between
the state of something that is uncertain (e.g. an electron in a probability cloud, mass indamped oscillations), ANDthe state of something having a definite value (e.g. having a definite position, a definitemomentum, a definite measured value, and a definite time of occurrence)
When an object can definitely be “pinned down” in some respect, it is said to
possess an eigenstate.
The word eigenstate is derived from the German/Dutch word eigen, meaning
inherent or characteristic.
An eigenstate is the measured state of some object possessing quantifiable
characteristics such as position, momentum, etc. The state being measured and
described must be observable (i.e. something such as position or momentum that
can be experimentally measured either directly or indirectly), and must have a
definite value, called an eigenvalue.
Source: http://en.wikipedia.org/wiki/Introdu tion_to_eigenstatesibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 3 / 31
Introduction
Eigenvalues play an important role
in situations where the matrix is a
transformation from one vector
space onto itself.
Systems of linear ordinary
differential equations are the
primary examples.
The eigenvalues can correspond to
critical values of stability
parameters, or energy levels of
atoms, or different characteristic
polynomials (of eigenvalue)
describing geometric
transformations (scaling, unequal
scaling, rotation, horizontal shear,
hyperbolic rotation) or frequencies
of vibration–see Figure 1.
Figure 1: Vibration of mass-spring systemwith three degress of freedom.
The equations of motion for a system of masses andspring shown in Figure 1 are
m1q̈1 + (k1 + k2 + k4)q1 − k2q2 − k4q3 = 0
m2 q̈2 − k2q1 + (k2 + k3)q2 − k3q3 = 0
m3 q̈3 − k4q1 − k3q2 + (k3 + k4)q3 = 0(1)ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 4 / 31
Engineering ApplicationsExample 1
Figure 2: Forging hammer.ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 5 / 31
Engineering ApplicationsExample 1
Problem Statement:
A forging hammer of mass m1 is mounted on a concrete foundation block of mass m2.
The stiffness of the springs underneath the forging hammer and the foundation block
are given by k2 and k1, respectively (see Figure 2). The system undergoes simple
harmonic motion at one of its natural frequencies ω, which are given by
»
k1 + k2 −k2
−k2 k2
–
x1
x2
ff
= ω2
»
m1 0
0 m2
–
x1
x2
ff
(a)
where ω2 is the eigenvalue and ~XT = {x1 x2} is the eigenvector or mode shape
(displacement pattern) of the system. Determine the natural frequencies and mode
shapes of the system for the following data:
Solution:
Work through the example.
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 6 / 31
Engineering ApplicationsExample 2
Problem Statement:
The pin-ended column shown in
Figure 3(a) is subjected to a compressive
(axial) force, P. If the column is
perturbed slightly as shown in
Figure 3(b), as might occur from a slight
vibration of the supports, it may not
return to horizontal position even after
removal of the distrubances; instead, the
deflection might grow if the load is
sufficiently large. Such load is called the
buckling load.
Solution:
Work through the example. Figure 3: Pin-ended column.
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 7 / 31
Definitions and Basic Facts
A matrix is called symmetric if it is equal to its transpose, i.e.
A = AT
or aij = aji (2)
Example:
A =
2
4
2 7 3
7 9 4
3 4 7
3
5 (3)
A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries
that is equal to its own conjugate transpose, A∗–i.e. the element in the i-th row and
j-th column is equal to the complex conjugate of the element in the j-th row and
i-th column, for all indices i and j, i.e.
A = AT = A∗
or aij = aji (4)
Example:
A =
2
4
2 (2 + i) 4
(2 − i) 3 i
4 −i 1
3
5ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 8 / 31
Definitions and Basic Facts
A matrix is termed orthogonal if its transpose equals its inverse, i.e.
AT = A
−1which entails A
T · A−1 = A
−1 · AT = 1 (5)
Example: A 2 × 2 rotation matrix, say for θ = 16.26◦,
A =
»
cos θ − sin θ
sin θ cos θ
–
A complex square matrix A is normal if
A∗ · A = A · A
∗
(6)
where A∗ is the conjugate transpose of A, e.g.
A =
2
4
1 1 0
0 1 1
1 0 1
3
5 (7)
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 9 / 31
Definitions and Basic FactsEigenvalue & Eigenvector
An eigenvalue and eigenvector of a square matrix [A] are a scalar λ, and a nonzero
vector ~X so that
[A]~X = λ~X
Consider matrix A and three vectors x1, x2, and x3,
A =
2
4
1 2 16 −1 0
−1 −2 −1
3
5 (8)
x1 =
2
4
16
−13
3
5 x2 =
2
4
−121
3
5 x3 =
2
4
23
−2
3
5 (9)
If we pre-multiply each of these vectors with matrix A we get
Ax1 =
2
4
000
3
5 Ax2 =
2
4
4−8−4
3
5 Ax3 =
2
4
69
−6
3
5 (10)
in other words, we have
Ax1 = 0x1 Ax2 = −4x2 Ax3 = 3x3
In this case we say 0, -4 and 3 are eigenvalues of the matrix A, and x1, x2 and x3 areeigenvectors of A.ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 10 / 31
Definitions and Basic FactsEigenvalue & Eigenvector
To find eigenvalues and eigenvectors of a given matrix we proceed as follows:
Form the matrix A − λI, i.e. substract λ from each diagonal element of A.
Solve the characteristic equation |A − λI| = 0 for λ.
Take each value of λ in turn, substitute into . . .
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 11 / 31
Definitions and Basic FactsEigenvalue & Eigenvector
Standard eigenvalue problem is defined by the homogeneous equations
(a11 − λ)x1 + a12x2 + a13x3 + . . . + a1nxn = 0
a21x1 + (a22 − λ)x2 + a23x3 + . . . + a2nxn = 0
. . .
an1x1 + an2x2 + an3x3 + . . . + (ann − λ)xn = 0 (11)
and in matrix form
[A]~X = λ~X (12)
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 12 / 31
Definitions and Basic FactsEigenvalue & Eigenvector
In Eq. (12),
1 [A] is a known square, (n × n),matrix
[A] =
2
6
4
a11 a12 . . . a1n
a21 a22 . . . a2n
. . .
an1 an2 . . . ann
3
7
5
(13)
2 ~X is the unknown n-componentvector,
~X =
8
>
<
>
:
x1
x2
. . .
xn
9
>
=
>
;
(14)
3 λ is an unknown scalar, a.k.a.eigenvalue.
Eq. (12) can be re-written as
[[A] − λ[I]]~X = ~0 (15)
where [I] is the identity matrix of
order n.
Eq. (15) has nontrivial solution
because it represents a system of n
homogeneous equation in n + 1
unknowns. Thus the determinant of
the coefficient matrix of ~X in
Eq. (15) must be zero:
|[A] − λ[I]| = 0 (16)
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 13 / 31
Definitions and Basic FactsCharacteristic Equation
If we expand Eq. (16) we get
˛
˛
˛
˛
˛
˛
˛
˛
(a11 − λ) a12 . . . a1n
a21 (a22 − λ) . . . a2n
. . .
an1 an2 . . . (ann − λ)
˛
˛
˛
˛
˛
˛
˛
˛
= 0 (17)
which, upon further expansion, gives an nth order polynomial in λ,
Pn(λ) = anλn + an−1λ
n−1 + . . . + a2λ2 + a1λ + a0 = 0 (18)
called the characteristic equation, where a0, a1, a2 . . . an are coefficients of
polynomial. Assuming λ1, λ2, . . . , λn are roots of the characteristic equation, then
there will be n solutions to Eq. (12).
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 14 / 31
Definitions and Basic FactsCharacteristic Equation
Corresponding to each distinct eigenvalue λi, a nontrivial solution of linear
equations in Eq. (12) can be determined through
[A]~X(i) = λi~X(i) (19)
where
~X(i) =
8
>
>
>
<
>
>
>
:
x(i)1
x(i)2
. . .
x(i)n
9
>
>
>
=
>
>
>
;
(20)
is called the eigenvector corresponding to eigenvalue λi.
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 15 / 31
Definitions and Basic FactsStandard & General Eigenvalue Problems
Standard Eigenvalue Problem
Eigenvalue problem as expressed by Eq. (12),repeated below,
(a11 − λ)x1 + a12x2 + . . . + a1nxn = 0
a21x1 + (a22 − λ)x2 + . . . + a2nxn = 0
. . .
an1x1 + an2x2 + . . . + (ann − λ)xn = 0
and in matrix form
[A]~X = λ~X
is known as the standard eigenvalue problem.
General Eigenvalue Problem
Many physical problems, however, areexpressed as the general eigenvalue problemgiven by
[A]~X = λ[B]~X (21)
General eigenvalue problem of Eq. (21) can bereduced to the standard eigenvalue problem ofEq. (12). We shall deal with this conversionlater!
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 16 / 31
Definitions and Basic FactsExample 1: Finding Eigenvalues & Eigenvectors of a 2 × 2 Matrix
If
A =
»
0 1−2 −3
–
then the characteristic equation is
|A − λ · I| =
˛
˛
˛
˛
»
0 1−3 −3
–
−
»
λ 00 λ
–˛
˛
˛
˛
= 0
˛
˛
˛
˛
»
−λ 1−2 −3 − λ
–˛
˛
˛
˛
= λ2 + 3λ − 2 = 0
and the two eigenvalues are
λ1 = −1, λ2 = −2
Eigenvector, v1, associated with the eigenvalue, λ1 = −1, is
A · v1 = λ1 · v1
(A − λ1) · v1 = 0»
−λ1 1−2 −3 − λ1
–
· v1 = 0
»
1 1−2 −2
–
· v1 =
»
1 1−2 −2
–
·
»
v1,1
v1,2
–
= 0ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 17 / 31
Power Method
Since each eigenvector is associated with an eigenvalue, we often refer to an
eigenvector X and eigenvalue λ that correspond to one another as an eigenpair.
Many of the “real world” applications are primarily interested in the dominant
eigenpair.
The dominant eigenvector of a matrix is an eigenvector corresponding to the
eigenvalue of largest magnitude (for real numbers, largest absolute value) of that
matrix. The method that is used to find this eigenvector is called the power
method.
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 18 / 31
Power MethodSequencing the Iteration
Like the Jacobi and Gauss-Seidel methods, the power method for approximating
eigenvalues is iterative.
1 First we assume that the matrix A has a dominant eigenvalue with corresponding
dominant eigenvectors.
2 Then we choose an initial approximation of one of the dominant eigenvectors of A.
This initial approximation must be a nonzero vector in Rn.
3 Finally we form the sequence given by
x1 = A x0
x2 = A x1 = A(A x0) = A2x0
x3 = A x2 = A(A2x0) = A
3x0
...
xk = A xk−1 = A(Ak−1x0) = A
kx0 (22)ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 19 / 31
Power MethodSequencing the Iteration–Example 1
We demonstrate those sequences on the matrix
A =
»
3 61 4
–
and, for Step 1, arbitrarily choose
x0 =
»
11
–
x1 = A x0 =
»
3 61 4
– »
11
–
=
»
95
–
=⇒ x′
1 =
»
1.81
–
x2 = A x1 =
»
3 61 4
– »
1.81
–
=
»
11.45.8
–
=⇒ x′
2 =
»
1.9655172411
–
x3 = A x2 =
»
3 61 4
– »
1.9655172411
–
=
»
11.896551725.96551724
–
=⇒ x′
3 =
»
1.9942196531
–
x4 = A x2 =
»
3 61 4
– »
1.9942196531
–
=
»
11.982658965.99421965
–
=⇒ x′
4 =
»
1.999035681
–
It looks like an eigenvector is
x =
»
21
–
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 20 / 31
Power MethodSequencing the Iteration–Example 1 (continued)
The corresponding eigenvalue is
xTAx
xTx=
ˆ
2 1˜
»
3 61 4
– »
21
–
ˆ
2 1˜
»
21
– =[30]
[5]=⇒ λ = 6
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 21 / 31
Power MethodAlgorithm & Matlab Implementation
The power method iteration in effect
takes successively higher powers of
matrix times initial starting vector
Algorithm: Power Method
x0 = arbitrary nonzero vector
for k = 1, 2, . . .
yk = A xk−1
xk = yk/||yk||∞
end
Implementing the algorithm in Matlab:
Matlab Session>> A = hilb(5) % matrix A>> x0 = ones(5,1) % nonzero ve tor>> yk = A*x0>> xk = yk/norm(yk)Compare the new value of x with the original.
Repeat the last two lines (hint: use the scrollup button).
Compare the newest value of x with theprevious one and the original. Notice thatthere is less change between the second two.
Repeat the last two commands over and overuntil the values stop changing.
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 22 / 31
Faddeev-Leverrier Method
Let A be and n × n matrix. The determination of eigenvalues and eigenvectors
requires the solution of
AX = λX
where λ is the eigenvalue corresponding the eigenvactor X. The values λ must
satisfy the equation
det(A − λI) = 0
Hence λ is a root of an nth degree polynomial P(λ) = det(A − λI), which we write
in the form
P(λ) = λn + c1λ
n−1 + c2λn−2 + . . . + cn−2λ
2 + cn−1λ + cn (23)
The Faddeev-Leverrier algorithm is an efficient method for finding the coefficients
ck of the polynomial P(λ). As an additional benefit, the inverse matrix A−1 is
obtained at no extra computational expense.ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 23 / 31
Faddeev-Leverrier MethodTrace of Matrix A
The trace of the matrix A, written Tr[A], is
Tr[A] = a1,1 + a2,2 + . . . + an,n (24)
The algorithm generates a sequence of matrices Bnk=1 and uses their traces to
compute the coefficients of P(λ),
B1 = A and p1 = Tr[B1]
B2 = A(B1 − p1I) and p2 =1
2Tr[B2]
... . . ....
Bk = A(Bk−1 − pk−1I) and pk =1
kTr[Bk]
... . . ....
Bn = A(Bn−1 − pn−1I) and pn =1
nTr[Bn] (25)ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 24 / 31
Faddeev-Leverrier MethodCharacteristic Polynomial
Then the characteristic polynomial, Eq. 23, can be re-written as
P(λ) = λn + p1λ
n−1 + p2λn−2 + . . . + pn−2λ
2 + pn−1λ + pn (26)
and the inverse matrix is given by
A−1 =
1
pn(Bn−1 − pn−1I) (27)
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 25 / 31
Faddeev-Leverrier MethodExample 1
Problem Statement:
Use Faddeev’s method to find the characteristic polynomial and inverse of the matrix
2
4
2 −1 1
−1 2 1
1 −1 2
3
5
Solution:
A =
2
4
2 −1 1
−1 2 1
1 −1 2
3
5
B1 =
2
4
2 −1 1
−1 2 1
1 −1 2
3
5
p1 = Tr[B1] = 6ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 26 / 31
Faddeev-Leverrier MethodExample 1
Solution (continued):
B2 = A(B1 − p1I)
B2 =
2
4
2 −1 1
−1 2 1
1 −1 2
3
5
0
@
2
4
2 −1 1
−1 2 1
1 −1 2
3
5 − 6
2
4
1 0 0
0 1 0
0 0 1
3
5
1
A
=
2
4
−6 1 −3
3 −8 −3
−1 1 −8
3
5
p2 =1
2Tr[B2] =
1
2(−22) = −11
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 27 / 31
Faddeev-Leverrier MethodExample 1
Solution (continued):
B3 = A(B2 − p2I)
B3 =
2
4
2 −1 1
−1 2 1
1 −1 2
3
5
0
@
2
4
−6 1 −3
3 −8 −3
−1 1 −8
3
5 − (−11)
2
4
1 0 0
0 1 0
0 0 1
3
5
1
A
=
2
4
6 0 0
0 6 0
0 0 6
3
5
p3 =1
3Tr[B3] =
1
3(18) = 6
The characteristic polynomial is thus
P(λ) = λ3 −
3X
i=1
piλn−i = −6 + 11λ− 6λ
2 + λ3ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 28 / 31
Faddeev-Leverrier MethodExample 1
Solution (continued):
The inverse matrix is
A−1 =
1
p3(B3−1 − p3−1I)
=1
6
0
@
2
4
−6 1 −3
3 −8 −3
−1 1 −8
3
5 − (−11)
2
4
1 0 0
0 1 0
0 0 1
3
5
1
A
=1
6
0
@
2
4
−6 1 −3
3 −8 −3
−1 1 −8
3
5 −
2
4
−11 0 0
0 −11 0
0 0 −11
3
5
1
A
=1
6
2
4
5 1 −3
3 3 −3
−1 1 3
3
5 =
2
6
6
4
56
16
− 12
12
12
− 12
− 16
16
12
3
7
7
5
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 29 / 31
Matlab eig FunctionExample 1
Problem Statement:
2
4
−149 −50 −154
537 180 546
−27 −9 −25
3
5
8
<
:
x1
x2
x3
9
=
;
=
2
4
λ1 0 0
0 λ2 0
0 0 λ3
3
5
8
<
:
x1
x2
x3
9
=
;
Solution:
Matlab Session>> A = [-149 -50 -154; 537 180 546; -27 -9 -25℄>> [X,lambda℄ = eig(A)>> lambda1 = lambda(1,:)>> lambda2 = lambda(2,:)>> lambda3 = lambda(3,:)>> X1 = X(:,1)>> X2 = X(:,2)>> X3 = X(:,3)>> LHS1 = A*X1>> RHS1 = lambda1*X1>> % Che k if LHS=RHSibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 30 / 31
Bibliography
1 STEVEN C. CHAPRA & RAYMOND P. CANALE (2009): Numerical Methods for Engineers, 6ed,ISBN 0-39-095080-7, McGraw-Hill
2 SINGIRESU S. RAO (2002): Applied Numerical Methods for Engineers and Scientists, ISBN0-13-089480-X, Prentice Hall
3 DAVID KINCAID & WARD CHENEY (1991): Numerical Analysis: Mathematics of ScientificComputing, ISBN 0-534-13014-3, Brooks/Cole Publishing Co.
4 STEVEN C. CHAPRA (2012): Applied Numerical Methods with MATLAB for Engineers andScientists, 3ed, ISBN 978-0-07-340110-2, McGraw-Hill
5 JOHN H. MATHEWS & KURTIS D. FINK (2004): Numerical Methods Using Matlab, 4ed, ISBN0-13-065248-2, Prentice Hall
6 WILLIAM J. PALM III (2011): Introduction to MATLAB for Engineers, 3ed, ISBN978-0-07-353487-9, McGraw-Hill
ibn.abdullah�dev.null 2014 SKMM 3023 Applied Numerical Methods Matrix Eigenproblems 31 / 31
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