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1
Lecture 2: Eigenvalue-Eigenvector and Orthogonal Matrix (Part B
Chapter 7 in Advanced Eng. Math. By E. Kreyszig)
Definition For a given square matrix A, if equation AX = X, with
is a nonzero number and X is a nonzero vector exists then is the
eigenvalue and X the eigenvector of A. Both and X are derived from
A, they represent certain characteristics of A. The definition
shows that, square matrix A multiplying its eigenvector only
changes the eigenvectors magnitude with a proportional factor of
(eigenvalue). Calculation
.3 ;5 3,53 45 try 045210)(22)2)(2)(6()1)(1)(3(
)3)(2(2)1)(6(2))(1)(2(
0-02-1-6--123-2-2-
0100010001
02-1-6-123-22-
02-1-6-123-22-
0 |I| from Find (1)
steps Two 0 |I-| requests RuleCramer the,for solution nontrivial
has LES shomogeneou theIf
)(
32123
3
2
1
3
2
1
111
=====++
=
++
=
=
=
=
=
====
xxx
xxx
A-
AX
OXIAOIXAXXIAXXXA nnnnnnn
Roots of a polynomial: the number of roots equals the polynomial
order and the product of all roots equals the polynomial constant.
(2) Find eigenvector X from homogeneous LES, 0)( = XIA with the
calculated using GE a) For = 5
=
=
=
=
=
=
=+
=++
121
20002052
000210521
16-8-016-805-2-1-
32-16-016-805-2-1-
3-27-6-425-2-1-
5-2-1-6-423-27-
5-02-1-6-5-123-25-2-
1
13
12
11
132
321
ccxcxcx
Xxxxxx
Notice: there are 2 Equations including 3 unknown components of
the eigenvector (x1, x2, x3) so 1 independent component x1 can take
any value, c1, and the other components, x2 and x3, have the values
corresponding to c1. In general, for a LES with n unknowns, after
Gauss Elimination, we have r = rank (A) being the number of
independent equations in the LES then, (n r) equals the number of
independent unknowns. The eigenvector needs to be represented as a
linear combination of the (n - r) independent vectors. b) For = -3
(the double roots)
+
=
=
=
+
=
=
=
=
==
=+
=
=
=
103
012
103
and 012
10
01
take Usually,23
r,eigrnvecto in the componentst independen as ; takeIf
0322
13
0000003-21
42-1-6-423-21
)3(-02-1-6-)3(-123-2)3(-2-
32323
2
33
22
23
3322
321
kkXXXkkxx
cxcxxx
X
cxcx
xxxrn
rn
Note: you can use any two independent vectors as
3
2
xx then calculate x1 and obtain the eigenvector X.
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2
Rules 1. Values of equal the roots of characteristic polynomial
derived from |A| = 0. If the order of the polynomial is k, you have
k real and/or complex roots, they may be distinct or multiple. A
square matrix, Ann can have at maximum n distinct roots
(eigenvalues). 2. All eigenvectors corresponding to distinct
eigenvalues are independent.
21 2
1
11
2
2
12 1 2
2
5 3 5 30 (5 ) 9 0 8, 2.
3 5 3 58,
5 8 3 -3 3 1
3 5 8 0 0 12,
5 2 3 3 3 1 and independent.
3 5 2 0 0 1
A
x X
x
xX X X
x
= = = = =
=
= =
=
= =
3. AT has the same eigenvalues as A. Example eigenvalues and
eigenvectors for 2d square matrices: Applications
1. Principal Directions in vector transformation (by multiplying
a matrix) A circle elastic membrane in the x1-x2 plane with
boundary equation: x12 + x22 = 1 is stretched so that any point P
(x1 x2) on the boundary goes to point Q (y1 y2) by = !! = = 5 33 5
!!
In the Principal Direction, points P moves to Q in the same
direction: 5 33 5 !! = !! 5 33 5 = 5 ! 9 = 0 ! = 8, ! = 2
eigenvectors ! : 5 8 33 5 8 !! = 0 ! ! = 0 1 independent ! = ! = c!
!! = c! 11 ! : 5 2 33 5 2 !! = 0 ! + ! = 0 1 independent ! = ! = d!
!! = d! 11
Illustration
Matrix
1011
Eq. 2 2+ 1 = 0 2 2k + k2 = 0 ( k1)( k2) = 0 2 2 cos + 1 = 0 1, 2
1= 2=1 1=2= k 1= k1 2= k2 1,2= cos isin
X1, 2 ( )01 ( ) 1) (0, 0 1, ( ) 1) (0, 0 1, (1, -i) (1, i)
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3
2. Discrete Time Markov Chain A random process of state vector
Xn1 characterized as memoryless: the next state Xt+1 depends only
on the current state Xt but not on the sequence of events that
preceded it: !!!!! = !!!!! , !!!!!!!!!!!! = 0.8 0.1 0.10.1 0.7 0.20
0.1 0.9
!! = 1!! = 0!! = 1 = !!!! = 0.9!!!! = 0.3!!!! = 0.9 !" =
probability of transforming from !! to !!!! !!!! = 0.9 includes 0.8
from !! ; 0.1 from !! and 0.1 from !! .
3. Vibration analysis
Two masses, m1 and m2, hang on two springs (k1 and k2) as shown.
With a small disturb, the system will vibrate around it
force-balanced positions y1 = 0 and y2 = 0. Calculate the
displacements of m1 and m2: y1(t) and y2(t).
From Newtons 2nd law and Hooks law.
221211112221
2
1 )( )( ykykkykyykdtydm ++==
221212222
2
2 )( ykykyykdtydm ==
=
+=
2
1
22
2
21
2
2
1
22
221
22
2
2
21
2
1
2225)(
yy
dtyddtyd
yy
kkkkk
dtydm
dtydm
and If
22
11
=
=t
t
exyexy
=
t
t
exex
dtyd
dtyd
2
2
21
22
2
21
2
Xxx
xx
xx
xx
exex
exex
t
t
t
t
r Eigenvecto ; Eigenvalue 22
2522
2522
25 22
1
2
12
2
12
2
21
2
12
2
21
=
=
=
y1
y2
m1 = 1 k2 = 2 m2 = 1
0
0
k1 = 3
-
4
=
=
==
=
=
=
+
+==
==
==
==
=
=
=
+
+===
===
=
=
ti
ti
itt
itt
ceycey
xcX
Oxx
Oxx
xx
i
ceexyceexyxcX
Oxx
Oxx
xx
i
xx
xx
62
61
22
2
1
2
1
2
12
22
1111
2
1
2
1
2
11
212
2
1
2
1
21
2
0021
4221
622
265 ;6 6For
221
0012
12
24
122215
;1 1For
;6;10-2-2
2-5-
2225
The system can vibrate in different modes! 2. Diagonalization of
a square matrix Ann with n independent eigenvectors (distinct
eigenvalues) Diagonalization means transforming a non-diagonal
matrix into an equivalent diagonal matrix, hence it becomes simple
for operations. If matrix Ann with distinct eigenvalues, it must
have independent eigenvectors. Form a new matrix P whose column
vectors are these distinct eigenvectors (P is called the modal
matrix of A). It can be shown that P-1 exists (|A| 0) and the
matrix product P-1AP = D. For this diagonal matrix D the diagonal
elements are the distinct eigenvalues. The order or the eigenvalues
in D is the same as the order of column vectors in P:
==
n
DAPP
...000.000...00...0
2
1
1
=
==
=
=
=
==
=
=
==
==
=
1005
1111
2332
1111
21
1111
21 and
1111
in order r eigenvecto theChange5001
1111
2332
1111
21
1111
21 and
1111
11
5 ;1
1 1
2332
Given
11
1
1222111
APPDPP
P
APPD
PPcXcXA
=
==
=
=
==
==
=
1006
1114
2145
and 11
141-1
1 ;14
62145
Given
54
51
51
51
1
54
51
51
51
12211
AXXD
XXXXA -
We can retrieve A from D and P: AIAIPPAPPPAPPPPDP ==== )()()(
11111 3. Matrix Power calculation using 1= PDPA
. 112112 === PPDAPPDPDPPDPA kk
=
==
=
=
=
5001
1111
21
5001
1111
then 1111
21 and
1111
2332
Given 23123231 PPDAPPA
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5
4. Similar Matrix Matrix C is formed by two given matrices A and
B with ABBC 1= then C and A are similar matrices and the process is
called a Similarity Transformation of A. Features of Similar Matrix
C and A share the same eigenvalues {i} and their eigenvectors have
the relation AC XBX
1= 1 25 52 15 5
1 25 51 12 15 5
5 41) Given matrix and test the similarity transformation.
1 2
Let and
Eigenvalues and eigenvectors of 5 4 5- 4
1 2 1 2-
T
X A
P X P X X
A
A
= =
= = = =
=
21 2
1
2
0 (5- )(2- ) 4 0
7 6 0 ( 6)( 1) 0 6, 1.Eigenvectors
65-6 4 -1 4 -1 4 41 2-6 1 -4 0 0 1
15-1 4 4 4 1 1 11 2-1 1 1 0 0 1
Similarity Matrix o
X
X
= =
+ = = = =
=
=
=
=
1 2 1 2 23 65 5 5 5 5 51 1 12 1 2 1 21 125 5
5 55 5 5 5
23 65 5
21 125 5
f
5 4 1 2 5 4 1 21 2 2 1 1 2 2 1
Eigenvalues and eigenvectors of
23 5 60 0
21 12 5
-
A
A P AP
A
= =
= =
%1
2
123 65 5
21 125 5
1 25 51 1 12 1 5 55 5
6 as
1
6
6 23 5 6 6 7 6 7 6 6121 12 5 6 21 18 0 0 756
4 1 2 4 6 Similarity!
1 2 1 1 7
s A
P X
=
=
=
=
= = =
5. Orthogonal Matrix Orthogonal Vectors Two vectors A1n and B1n
have the scalar product (dot) AB = 0. A1n (B1n)T = 0 Orthogonal
vectors must be independent, but independent vectors may not be
orthogonal. Perpendicular is limited to 3D, Orthogonal for nay
Dimensions. Orthogonal Matrix A square matrix has all column/row
vectors {ai} orthogonal. aiak= 0 j k Orthonormal Matrix (Normalize
the column/row vectors in the orthogonal matrix) A real square
matrix is orthogonal matrix, iff its column/row vectors: (a1,,an)
have:
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6
aiak=1length have vectorscolumn/row all othereach
vectorscolumn/rowdifferent
if 1 if 0
=
=
kjkj
Features of an orthonormal matrix: (1) AT = A1; (2) |A| = 1, (3)
Eigenvalue || =1, and (4) forming a unit perpendicular coordinate
system. Check agiven orthnormal matrix A
IAAAA TT =
=
=
=
32
31
32
32
32
31
31
32
32
32
32
31
31
32
32
32
31
32
32
31
32
32
32
31
31
32
32
;
32
32
31
31
32
32
32
31
32
Rule A symmetric matrix A can construct an orthonormal matrix X
using its normalized eigenvectors. Two steps: 1) calculate
eigenvectors (2) normalization the eigenvectors.
=+
=+
=
=====
=
51
52
21
525
1
21
2
12
12-
020021
4221
042201
21
020012
1224
542251
05
0504)4)(1(042
214221
xx
xx
A
1|| 5212)51(
04)51(0512
251 0
of s
1
matrix? orthogonalan 1001
0) product dot (vector lorthonorma0)(:Tests
2
51
52
52
51
51
52
52
51
51
52
52
51
51
52
52
51
1
51
52
52
51
51
52
52
51
=
==
=+=
=
=
=
=
==
==+
=
ii
X
X
XXXX
X
T
Specific matrices which can be transformed into an orthogonal
matrix: 1) A square matrix An n has n distinct eigenvalues can
construct an orthogonal matrix from its independent eigenvectors.
The column or row vectors in this orthogonal matrix construct a
basis for Rn (a Cartesian coordinate system). 2) An symmetric
matrix A can derive an orthonormal system from its eigenvectors and
form an orthogonal matrix (by normalization) whose column/row
vectors is a basis of Rn (a Cartesian coordinate system)