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Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
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Chapter 3. Theory of matrices
M
f
f
151
__________________________________________________________________________
λ
()
f
⎛⎞
1
⎜⎟
⎜⎟
⎜⎟
-1
AW W
()=
f
⎜⎟
⋅⋅
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
In view of (3.7.3) it remains to define a function of arbitrary Jordan block
and index m:
n
...
λ
()
f
p
J
()
f
k
1
...
J
()
f
k
q
. (3.7.3)
∈J
mn
of the order
where both I and
m .
Proposition 3.7.1
Let the Jordan block be represented in a form (3.7.4) and function
with its
and definition (3.7.5) coincides with Definition 3.7.1.
mm
N are matrices of the order n , and mN is the nilpotent matrix of the index
m
successive derivatives, then
1m −
⎛⎞
λ
( ) ...
f
⎜⎟
⎜⎟
′′′
λλ λ
() () ()
ff f
1! 2! ( 1)!
⎜⎟
J
()
f
m
⎜⎟
⎜⎟
=
⎜⎟
λ
() ...
f
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
=λ +JIN, (3.7.4)
be defined at λ along
(1)
m
−
−
m
−
m
′
λλ
() ()
ff
1! ( 2)!
(2)
−
m
, (3.7.5)
... ... ...
′
λ
()
()
f
λ
f
1!
λ
()
f
Proof
Suppose (later this supposition will be discarded) that
is analytic in a vicinity of λ , thus it
can be expanded into convergent Taylor’s series:
()
k
∞
() ( )
fz z
=−λ
∑
=
k
()
f
λ
!
k
0
k
. (3.7.6)
Now, by substituting into (3.7.6) instead of z the Jordan block
arrive at
()
k
∞
() ( )
f
JJI
=−λ
∑
mm
=
k
()
f
λ
!
k
0
k
, (3.7.7)
in the form (3.7.4), we
J
m
Chapter 3. Theory of matrices
M
p
p
p
152
__________________________________________________________________________
but, in the right-hand side of (3.7.7) we have
()
mm
Thus, in view of nilpotency of matrix
−λ =JIN
the series in the right-hand side of (3.7.7) becomes
N
m
. (3.7.8)
finite:
()
k
−
m
1
()
f
λ
k
It remains to note that
where
0 is the zero matrix having p rows and q columns, and
,
q
matrix of the rank
0I
⎛⎞
21
==
NN
⎜⎟
mm
⎜⎟
⎝⎠
2, 2 2
mnm m
−−+ −
00
nm nm nm m
−+ −+ −+ −
JN
()
f
∈N
mn
⎛⎞
=
N
⎜⎟
m
⎜⎟
⎝⎠
=
∑
mm
=
0
k
and
0I
1, 1 1
mnm m
−−+ −
00
1, 1 1, 1
nm nm nm m
−+ −+ −+ −
, and
2, 2 2, 2
!
k
,...,
. (3.7.9)
, (3.7.10)
I is the unity square
0 ... 1
⎛⎞
⎜⎟
m
−
...
⎜⎟
0
−−
1, 1
0
nn
⎜⎟
⎝⎠
(3.7.11)
Taking into account (3.7.10), (3.7.11), we arrive at (3.7.5). Now, in view of (3.7.9), it is
sufficient to demand existence of the first
successive derivatives of f.
1m −
Remark 3.7.2
As a function of the Jordan block defined, by applying decomposition (3.7.1) it can then be
constructed the function of any non-semisimple matrix.
3.7.2. Examples
Example 3.7.1
The exponent of the Jordan block of the index 2:
21
⎛⎞
exp
⎜⎟
02
⎝⎠
According to Proposition 3.7.1 that exponent is
⎛⎞
21
⎛⎞
exp
⎜⎟
02
⎝⎠
2
e
⎜⎟
==
⎜⎟
⎜⎟
0
⎝⎠
. (3.7.12)
2
e
1!
2
e
22
⎛⎞
ee
⎜⎟
⎜⎟
0
⎝⎠
. (3.7.13)
2
e
Chapter 3. Theory of matrices
153
__________________________________________________________________________
Remark 3.7.3
Of course, this and the subsequent results should coincide with the direct substituting of the
Jordan block (3.7.12) into the corresponding Taylor’s series, as suggested by Definition 3.7.1.
Example 3.7.2
Square root of the Jordan block
310
⎛⎞
Again, applying Proposition 3.7.1 yields
310
⎛⎞
⎜⎟
sqrt 3 1 = 3
⎜⎟
⎜⎟
⎝⎠
⎜⎟
sqrt 3 1
⎜⎟
⎜⎟
⎝⎠
⎛⎞
1/2
3
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
3
⎜⎟
⎜⎟
⎜⎟
⎝⎠
. (3.7.14)
3
1/2 3/2
−−
33
−
28
−
1/2
1/2
3
(3.7.15)
2
1/2
3
Remark 3.7.4
Direct verification shows that
Thus, the square root is constructed correctly.
Example 3.7.3
Square root for the Jordan block
Applying the constructing procedure outlined in Proposition 3.7.1, fails due to inexistent (at
) of the derivatives of the square root function. This means, that the square root of the
x =
0
matrix (3.7.17) cannot be constructed by the developed method and, actually the square root of
the Jordan block (3.7.17) does not exist.
1/2 3/2 1/2 3/2
⎛⎞⎛⎞
1/2 1/2
33
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎝⎠⎝⎠
−− −−
33 33
1/2 1/2
3331
−−
28 28
1/2 1/2
−−
33
⋅=
22
1/2 1/2
33
01
⎛⎞
sqrt
⎜⎟
⎝⎠
. (3.7.17)
0
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎝⎠
310
(3.7.16)
3
Chapter 3. Theory of matrices
154
__________________________________________________________________________
Bibliography to Chapter 3
AGAIAN S.S.
Hadamard Matrices and Their Applications. Springer-Verlag, Berlin. 1985. ISBN-13: 9783540160564
NDERSON T.W.
A
An Introduction to Multivariate Statistical Analysis. 2nd edition, John Wiley. New York. 2003.
ISBN-13: 978-0471360919
ARNETT S.
B
Matrices: Methods and Applications. Oxford University Press, Oxford. 1990. ISBN-13: 9780198596806
ARNETT S. and STOREY C.
B
Matrices in Stability Theory. Thomas Nelson & Sons Ltd. 1970. ISBN-13: 978-0177616174
ASILEVSKY A.
B
Applied Matrix Algebra in the Statistical Sciences. Dover Publ., New York. 2005. ISBN-13:
978-0486445380
ECKENBACH E.F. and BELLMAN R.
B
Inequalities. CRC Press. 1990. ISBN-13: 978-0824784881
ELITSKII G.R. and LYUBICH YU.I.
B
Matrix norms and their applications. Birkhauser, Bale, 1988. ISBN-13: 978-0817622206
BELLMAN R.
Introduction to Matrix Analysis. 2d. ed. McGraw-Hill. New York. 1970.
HATIA R.
B
Matrix Analysis. Springer-Verlag, Heidelberg, 1996. ISBN-13: 978-0387948461
JORCK A., PLEMMONS R.J. and SCHNEIDER H. (editors)
B
Large Scale Matrix Problems. North Holland, Amsterdam. 1981.
RONSON R.
B
Schaum's Outline of Theory and Problems of Matrix Operations. McGraw-Hill, New York.
1989. ISBN-13: 978-0070079786
IARLET P.
C
Introduction to numerical linear algebra and optimization. Cambridge texts in Applied
Mathematics. Cambridge University Press, Cambridge, 1989. ISBN-13: 978-0521339841
HATELIN F.
C
Eigenvalues of Matrices. John Wiley & Sons, N.Y. 1993. ISBN-13: 978-0471935384
OLLAR A.R. and SIMPSON A.
C
Matrices and Engineering Dynamics. Ellis Harwood, Chichester. 1987. ISBN-13: 9780853128519
ULLEN C.G.
C
Chapter 3. Theory of matrices
155
__________________________________________________________________________
Matrices and Linear Transformations. Dover Publications. 1990. ISBN-13: 978-0486663289
E
RICKSEN J.L.
Tesnsor fields. In: Handbook der Physik. Bd. III/1. Berlin, Springer Verlag. 1960. pp. 794-858.
Zbl 0118.39702
RANKLIN J.
F
Matrix Theory. Prentice-Hall, Englewood Cliffs, N.J. 1968.
ANTMACHER F.R.
G
Applications of the Theory of Matrices. Dover Publications, 2005. ISBN-13: 978-0486445540
UTKEPOHL H.
L
Handbook of Matrices. John Wiley & Sons, Chichester. 1996. ISBN-13: 978-0471966883
ARCUS M. and MINC H.
M
A Survey of Matrix Theory and Matrix Inequalities. Dover Publications, 2010. ISBN-13: 9780486671024
EYER C.D.
M
Matrix analysis and applied linear algebra. SIAM, N.Y., 2000. ISBN-13: 978-0898714548
TEWART G.W.
S
Matrix Algorithms. Vol. II: Eigensystems. SIAM, N.Y., 2001. ISBN-13: 978-0898715033
ILKINSON J.H.
W
The Algebraic Eigenvalue Problem. Oxford University Press, N.Y., 1988. ISBN-13: 9780198534181
Chapter 4.
157
Equation Chapter 4 Section 0
Ordinary differential equations
In this chapter we present both formal description of the basics of the theory of
ordinary differential equations (ODE) and give examples of solutions for differential
equations arising in different areas of mechanical engineering.
The following monographs can be regarded as a general source for references:
Coddington and Levinson (1984), Hale (2009), Hartman (1964), Polyanin and Zaitsev
(1995), Tenenbaum and Pollard (1985), and Zwillinger (1997). A different approach to
the theory of ordinary differential equations based on the concept of flows on
manifolds is adopted by Arnold (1989), (2006); see also Kambe (2009).
Primarily non-linear differential equations are studied in monographs by Drazin (1992),
McLachlan (1958), Melin and Castillo (2001), Reissig, Sansone, and Conti (1974),
Strogatz (2001).
Numerical methods for solving ordinary differential equations are discussed by Butcher
(2008), Hairer and Wanner (1996), Lambert (1991), and Zwillinger (1997).
Applications of ordinary differential equations to different problems in mechanics and
geophysics are considered in a lot of manuscripts and textbooks, among which the
following are worth to mention: Benest and Froeschle (1998), De Costa (2010), Inman
(2007), Roberts (2010), and Strogatz (2001). Several classic books on the theory of
vibrations may be of special interest to the reader: Den Hartog (2008), Piersol and Paez
(2009), Timoshenko (2008), and a more theoretical work by Andronov, Vitt, and
Khaikin (1987).
Chapter 4. Ordinary differential equations
x
x
p
158
__________________________________________________________________________
4.1. Equation Chapter 4 Section 1 Basic
concepts
4.1.1. Basic definitions
Definition 4.1.1 (Ordinary differential equation; ODE)
Ordinary differential equation (ODE) of the order
where
F
successive derivatives
“external loading”
except its non-degeneracy with respect to the variable
the ODE of the
Remark 4.1.1
Sometimes in the definition for ODE the external loading ()
In such a case the implicit function defining the ODE is written as
Examples 4.1.1
is a function of
′
( ),..., ( )
()pt . At this moment we do not impose any restrictions on the function
-th order.
n
n is an equation:
()
( ; , ,..., , ) 0
Ftxx x p′= , (4.1.1)
variables: the independent variable
3n +
()
txt
n
n
, a function
t
with respect to the independent variable
()n
. The latter ensures Eq. (4.1.1) to be
, its
()xt
, and the
t
,
F
t is moved to the right-hand side.
()
n
′
( ; , ,..., ) ( )
Ftxx x pt
, (4.1.2)
=
Herein we give examples of some of the ODEs that can be often found in applications
A linear ODE of free small oscillations of a mathematical pendulum (equation of harmonic
1)
oscillator
where
Another example of linear ODE, this time describing free small oscillations of a
2)
mathematical pendulum with friction (or equation of damped oscillations)
where
)
2
is the circular frequency of small oscillations, called also the natural frequency.
ω
20xrx x++ω=
r
is the viscous force per unit speed.
(4.1.3)
0xx+ω =
2
(4.1.4)
Chapter 4. Ordinary differential equations
159
__________________________________________________________________________
A nonlinear ODE of large oscillations of the pendulum (sometimes this is called physical
3)
pendulum)
2
sin 0xx+ω = (4.1.5)
this time parameter
ω does not define the natural frequency. Despite its simplicity an
analytical solution of Eq. (4.1.5) is known only in terms of elliptical integrals, while for the
physical pendulum with linear viscous friction
2sin0xrx x++ω =
2
(4.1.6)
no analytical solutions are known.
4)
The following equation gives another example of a frequently found nonlinear ODE,
describing small oscillations of a system with
2() 0xrsignx x+⋅ +ω=
dry (called also Coulomb) friction
2
, (4.1.7)
or in the equivalent form
x
Yet another nonlinear ODE that corresponds to a spring element with non-linear response
5)
20
xr x
+⋅+ω=
20xrx xkx++ω+ =
2
x
23
()
. (4.1.8)
. (4.1.9)
This equation, known also as the Duffing equation with linear viscous friction, describes
non-linear oscillations of an element containing spring with cubic nonlinearity. At some
values of parameters r, ω, and k the equation can lead to chaotic solutions.
Definition 4.1.2 (Autonomous ODE)
An ordinary differential equation (4.1.1) is called autonomous, if it does not explicitly contain
the independent variable t; thus the autonomous differential equation is defined by the
following implicit function:
Remark 4.1.2
According to the previous definition, the autonomous differential equation should necessary be
either homogeneous, or with very simple constant inhomogeneity.
Examples 4.1.2
a) According to Definition 4.1.2 the following equation is autonomous:
b) At the same time, equation
coefficients depends upon
F x x x const
''' sin( ) 2xx t x+= is non-autonomous, since one of its
.
t
=
2
()
n
′
(, ,..., )
(4.1.10)
2
''' 2xx x+=
,
Chapter 4. Ordinary differential equations
x
x
x
x
160
__________________________________________________________________________
Definition 4.1.3 (Initial value problem; Cauchy problem)
The initial value or Cauchy problem is a problem of finding the solution of Eq. (4.1.1)
satisfying the initial conditions at some
( ) , ( ) ,....., ( )
taxta x ta
== = (4.1.11)
00 01 0 1
′
Definition 4.1.4 (Sturm-Liouville problem; two-point boundary value problem)
For ODE of the second order along with the initial condition, another type of (boundary)
conditions can be formulated:
tt=
:
0
(1)
n
−
n
−
() , ()
taxta==
00 11
Condition (4.1.12) is referred to as Sturm-Liouville or the two-point boundary value problem.
Definition 4.1.5 (General solution of ODE)
Solutions (there can be infinite number of them) of the homogeneous Eq. (4.1.2), i.e. with the
zero right-hand side, are called the general solutions.
Definition 4.1.6 (Partial solution of ODE)
Any solution of the inhomogeneous Eq. (4.1.2), i.e. with a nonzero right-hand side, is called as
the partial solution.
Proposition 4.1.1
Any ODE of the n-th order can be reduced to a system of n ODEs of the first order.
Proof
Denoting
() ()
txt≡
0
and introducing 1n − new functions
(4.1.12)
() ()
xt x t
10
() ()
xt x t
21
′
=
′
=
(4.1.13)
...........
tx t
() ()
nn
−−
12
′
=
yield Eq. (4.1.1) in the form
(; , ,... , ) ()
Ftx x x x pt
01 1 1
nn
−−
′
=
Equations (4.1.13), (4.1.14) form the desired system of
. (4.1.14)
n ODEs of the first order:
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