In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system

Sartain, R. L.

English

NASA Technical Reports Server

i N86-14099
Transformation Matrices " - -
Between
, Non-Linear and Linear
Differential Equations
>-., ) Robert L. Sartain
Associate Professor
Department of Mathematics and Physics
" " Houston Baptist University
_. Houston, Texas
.._.
ABSTRACT
- In the linearization of systems of non-linear differential equations, we consider those
._. systems which can be exactly transformed into the second order linear differential
" equation YJI-AYI-BY=0 where Y, Y_ and yUare n x l vectors and A and B are constant
" n x n matrices of real numbers. We use the 2n x 2n matrix M= to transform the
; above matrix equation into the first order matrix equation X I = MX. We study _,.
specilically the matrix M and the conditions which will diagonalize or triangularize M.
We indicate transformation matrices P and P'! to accomplish this diagonalization or
triangularization and how to use these to return to the solution of the second order
matrix differential equation system from, the first order system.
We conclude with a study of the relationship between the diagonalization of M to that
of the submatrices A and B.
Center Research Advisor: Victor R. Bond
- _ 22-1
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1986004609-328
https://ntrs.nasa.gov/search.jsp?R=19860004630 2020-03-20T16:33:05+00:00Z
Introduction
(Szebehely, 1976), (Bond and Horn9 197g), and (Bond, 1982) have studied the
linearization of systems of non-linear differential equations. In each of these cases
the transformation of one system to the other involves square matrices A _ B of
' : order n.
In the present paper we consider only those non-linear differential equation systems
which cap be exactly transformed into the second order linear differential equation
_ system Y"-AYI-BY=0 where the matrices A and B are constant matrices of real
numbers and where Y, Y_, and Y_'are n x ! vectors.
We form the matrix M =I; "_Jwhichis a matrix of order 2n which arises in the standard
-: transformation of the second-order linear equation to a first-order linear equation by
the transformation
Ze-- Y
whereY_ thev_tor (Y,,Y_,...,Yn)_ndY': (Y/, Y_,...,%)rI
z,---Ylj _,
i yi i ,IObserve that this transformation yields Zp = = Za and Zz = Y = AZ. + BZ_.
v/ritten in matrix form we have
z;J- " z,
LettingX =Iz:]andM = [;Altheeq uati°n Yu= AYl+ BY can be reduced l:o the linear
first-order matrix equation X_= MX,
This research is concerned with the following questions:
(1.) V/hat are the conditions on the matrix M which will allow us to solve the matrix
'.
differential equation system X ° = MX.
,=,
::! 22-2
+3
198(5004609-329
i.
' (2.) Under the assumption that this system can be solved, how does one actually
:' recover the matrices to return to the original second order differential equation
• systemYH AYt_BY=O.
r
This paper shall try to answer these questions as completely as possible.
v
.!
,Jl
22-3
1986004609-330
Theory
/
The solution of the first-order linear differentia! equation system X = MX depends
heavily on the matr_,x M given. We consider two cases. The most satisfactory solution
occurs when the matrix M is similar to a diagonal matrix and much of our research has
centered _)on what must be known about the submatrices A and B of M =F__1in order
to assure that M can be diagonalized. We give some elementary linear algebra theory
to familiarize everyone with the ideas.
1
For a linear homogeneous transformation such as M above, there exist scalars _and
" vectors X. which satisfy the equation MX_ : _ X_ • The values of _ for which this
equation is satisfied are called the eigenvalues of M and the vectors X.t which are fixed
for each _ are called the eigenvectors. There are manyunder the transformation M
other names associated with these values and vectors, some of which are
characteristic values and vectors, latent values and vectors, arid proper values and
.-.
: vectors.
Clearly, the zero vector will always satisfy the equation MX = _X for any _ chosen.
However, we desire to find nontrivial solutions to the problem. Rewriting the equation "_
as (M-)_I) X = 0 we see that any nontrivial vector X will satisfy the equation if and or ,y
if det (M-_I) = 0 where det stands for the determinant of the matrix M-_I.
The determinant of this is a polynomial equation in _. Since, in our case, M is of order
2n, the polynomial equation f(_) = 0 will be of degree 2n, and hence there _ill be 2n
: eigenvalues associated with M. These values may or may not be distinct, and we shall
discussthe consequences later in the paper.
,r,
iv,,
-- , 22.-,_,
t
]986004609-33]
d• For each distinct eigenvalue,_, satisfying f (A)= det (M-hi) = 0 there will exist at least
_ one non-trivial eigenvector X satisfying (M-_I) X = 0. (Stein, 1967).
The eigenvalues and eigenvectors play a central role in the solution of the differential?
y.
equation X I= MX and this is our purpose in discussing them here.
. A matrix M is said to be similar to the matrix C if there exist a nonsingular matrix P
. such that M = P CP -_.
i The matrix M is diagonalizable if it is similar to a diagonal matrix
D=Lo,..jr
where the large zeros indicate that all elements off the main diagonal are zeros.
i
Supposenow that the matrix M of our system X S = MX is known to be diagonalizable,
then there exists a nonsingular matrix P such that P_MP=D (AI, ..., _. ) where the
notation D (),a, ..., Aa_ ) will always mean a diagonal matrix where the _j, Aa, ..., A_._
are the eigenvalues of M and they lie on the main diagonal of D. See Theorems 6.7.1
and 6.8.1 of (Stein, 1967) for a proof of the above statement. _.
/
' If we now :set X = PZ where P is the matrix which diagonalizes M, then X _ = PZeand Z
= P-IX. So PZ t=Xl= MX or Z I= P'_MX = (P-IMP) Z Thus Zs = DZ where D is our
, diagonal matrix. Using elementary differential equation theory, it is known that a
solution of Z s = DZ is
" [ i l't
_. Cll
i [o'.,,
t a
4 -:
22-5 )
I
1986004609-332
So X = PZ will then solve the original system simply by multiplying the matrix P by the
vector 7.. Thus it is of paramount importance first • determine whether or not the
original matrix M can be dJagonalized and if it can, then how does one obtain the
matrix P which both diagonalizes M and gives the final solution for the X once we have
Z as shown above.
i Now consider the second case where the matrix M is not diagonalizable. Every square *
._. matrix M is similar to a triangular matrix T with the eigenvalues of M as the diagonal
3
. elements of T. A constructive proof of this fact is given in (Stein, 1967) as Theorem
_; 6.8.5. The process again generates a nonsingular matrix P such that P-IMP = T. Using
k
• the same construction of X = PZ given above, by elementary differential equation
-'- theory (Murdoch, 1957), one can solve the system for the original vector X : (X I , X_,
X;J T. If we assume the matrix T is upper triangular then after computing PZ, weem._
will have Xz** = C2_p., _'* _ as in the preceding case, and in general depending on the
multiplicity of the eigenvalues, i_ _ is an eigenvalue of multipl;city r, then the solution
for this case will be of the form eJt_(t ) where/o(t ) is a polynominal of degree r-I.
Thus a solution is possible even in,this case although not as easy to obtain or use. If
,, the _ )s are distinct then, for example, if X_,_ =Ca,,,=,_Cp then the solution for
X=,_-a = C_-j,=_. I -/- C__,_
where the c_._ 's are the nonzero elements in the upper triangular matrix which
correspond to each row i and column j position. One can use these solutions then to
s_'Jbstitute in and solve for the next X_ , and so on, until a complete solution is
obtained.
I
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1986004609-333
We next describe the process of actually obtaining P which will diagonalize or
: triangularize the matrix to give us the solutions indicated above. We consider each of
the casesseparately.
I
, Suppose first that M is diagonalizable, then there exists a set of 2n linearly
' independent eJgenvectors, say Uj, U=, .., U_and this is true regardless of whether or
not any eigenvalue has multiplicity greater than one or not. See (Edelen, 1976) for a
_ discussion of this property. If we supposethat UI corresponds to _l, U_ to/_=, ..., U_to ,_j_
_- even if some of the _ 's are equal, then the matrix P = (U_ , U=, ..., U_) where each
column of P is a vector of length 2n will be nonsingular since these 2n vectors are
J linearly independent. We observe that if _:=,_; for instance, there will still be distinct
p
eigenvectors UX and Ui which are linearly independent when M is diagonalizable. P
r nonsingular implies that it has an inverse and the product p-i MP will actually be a
djagona! matrix with the eigenvalues down the main ,'iagonal.
We will address the process of determining the eigenvectors for a given _ later in the "_ -
paper.
!
We now consider the problem of finding the nonsingular matrix P When the matrix M is
, no_._tdiagonalizable.
The process is given as follows. Process;
i (i.) Find an arbitrary eigenvector, X l for the first eigenvalue _l of your matrix.
" (We shall assume _l, _a_ ...,_are successively on the diagonal from top left toi
'r._ lower right.) I .£
" 22-7
1986004609-334
(2.) Form PI = (Xl,e_, ..., Ca,,. ) provided that P so formed is nonsingular.
(3.) Compute p-I and calculate the product pj-iMp I where M is our original
matrix.
(4.) Using _l MPI we now I_.ve _lin upper left corner. Call the submatrix formed
by crossing out the row and column which contains _l, M =. If MI is a 2x2 matrix
skip to step 10_...Otherwise compute an eigenvector for _. using M I (not M, but it ,
will be the same eigenvalue, _, as for M).
(5.) Form Pa = (X_ ,0j, ...,e_) where X_ is the 2n-I eigenvector found in step g.
(6.) Compute P_Jand calculate the product p_l M i Pa • This matrix is of order
=
2n-l.
- (7.) P_IM I PZ will now have _=. in the upper left hand corner and zeros in the
column belo, it.
2
(g.) Form the submatrix M._by crossing out the row and column containing ,_=.
t
(%) If M_ is 2x2 goto step tO__.If not continue the process as illustrated in steps
5-8 until a submatrix which is 2x2 is finally obtained.
(10.) When Mj is reached which is 2x2 the process will be changed as follows:
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1986004609-335
Process (Continued)
(a) Compute an eigenvector using M3 for _a__l.
(b) Compute an eigenvector using Mj for /_a,_-
(c) Let Pd'+l = (X_,.: Xj_). Note it does not contain standard basis vectors as in
• p
",e
previous cases.
' (11.) Finallyform the matrix P as follows:
p= Pl0 , ,.
: e3 0 z,=l, ,etc.
_. (12,) Compute p-i, Then the final product P-iMP will triangularize M and leave
" the eigenvalues on the diagonal of P-IMP. Note that this form will create an
,- upper triangular matrix.
i
Note: Each successive eigenvector is found by reducing Mj-)_I to canonical form
t
" for j = 09 ...,an-l. Then multiplying the reduced matrix by the appropriately sized
vector made up of the X; 's to be in the eigenvector. 5olve this system of linear
equations by ch(x_smg appropriate values for the arbitrary X's. You can choose X2 =
,2
1 for arbitrary X's if so desired. This then will make up the necessary eigenvectc_ for __,
, that _.
t
2
=
i,r
wj
1986004609-336
In determining eigcnvectors for an M which is diagonalizable, there are two cases to
consider. The first case is that for which al ) eigenvalues are distinct. In this case use
Gaussian elimination to reduce the matrix M -_I to canonical form (we use this name
to imply that the matrix has ones on the main diagonal and zeros off the main diagonal
in so far as possible and always zeros below the main diagonal). Other authors use the
" u t
phrase"row echelon form to describe this.
I
if C is the row echelon form of M -_I, then setting CX = O, where X is the 2n x 1
vector X = (X I , X z, ..., X_) r, will allow us to obtain the elements X I , X z, ..., Xa. of
_" the eigenvector for _ by solving the homogeneous system CX = O. Any convenient
". choice of the arbitrary X'swill give us an eigenvector for_.Since al! /_ 'sare distinct
- in this case, then each of the eigenvectors will be linearly independent so the matrix P
formed from the eigenvectors will be nonsingular.
. The second case is that where some of the eigenvalues have multiplicity greater than
one. In this case, if _is an elt'_nvalue of multiplicity _'; then tilere will be K;-linearly
independent eigenvectors associated with each such _. Recall that we are assuming
; the diagonalizability of M at this point, otherwise we would not know that this is
possible. For each i, one can obtain distinct eigenvectors by choosing different values
for the arbitrary variables in the equation CX = 0 when solved. Each of these
eigenvectors will not only be linearly independent of each other but also linearly
independent of eigenvectors obtained from distinct eigenvalues. See the proof of
Theorem 6.8.4 of (Stein, 1967) for a proof of this fact.
i
l Thus all 2n of the eigenvectors so obtained will be linearly independent and thus the
J
t m_ "rix P formed from the eigenvectors as previously indicated will be nonsingular.I
22-10
1986004609-337
We have made the assumption in the above work that M is diagonalizable. Consider
now the question of what conditions on M will guarantee that M is diagonalizable.
The most well-known theorem says that every real symmetric matrix is diagonalizable
, (Stein, 1967). See Theorem 6.$.6. However, our matrix M=[_ _1 is in generalnot
symmetric so we seek other criterion.
' A second theorem (6.8.2) indicates that M is diagonalizable if and only if M has 2n
linearly independent eigenvectors.
The difficulty with using this theorem is that we would like to determine whether or
not M is diagonal; :able before we compute the eigenvectors so we will know wt her
to use the diagonalization or triangularization process which we have described above
in finding the matrix P and its inverse.
< One further theorem (6.$.#.) tells us M is diagonaiizable if and only if for each '_i, the
multiplicity of _ is equal to k=2n-r where r _ the rank of M-J;I. (Stein, 1967).
/
Although this theorem is better than those above and works on all matrices, it still
" requires checking the multiplicity of every _" against the rank of M-_'I which could
be a rather formidable task.
P
Thus we look specifically at the form of our matrixM =[__] to determine, ii possible,
conditions on B and A which will help us decid _ the diagonalizability of M. To this
end, consider the following little known theorem from (Hohn, 1964).
22-11
I
1986004609-338
" [
,q
'i
IfPismxn ,Qismxn_Risnxmand Sisnxnandnonsingular, then
(let = detS-det (P-QfR). ._
J
We now assume thatP, Q, R, and S above are allsquareof sizen x n and alsothatSQ
= QS. W,=can thenprovethe followingtheorem.
det = det(PS-QR) I
Proof= Bythe,irsttheoremabove.detI: ;] =detS-det(P-QS"R). Thenusingthe '
well-known theorem that det AB = det A'det B when A and B are square n x _ *
matrices we apply this to the right member above to have det S" det (P-QS-' R) = det i
=:: (S(P-QS"R)) : det (SP-SQS'IR). Since we are assuming SQ = QS we have
det (SP-SQS"R) = det (SP-QSS" R) = det (SP-QR) since S'S" =I. F
commutes with every hlatrix of the same size, we have IA - AI so det M = IO'A - B'II=
det (-B) = (-I) _.det (B). i
F ,
Consideringthe eigenmatrixM-_I we have det(M-_I)= detLs
J
det ((-_IXA-_I)-B)= det(-_A.+_'I-B)= de'.(_21-_A-B). _,
Our purpose in looking at (let (M-_I) is to observe that (let (M- _, l) = 0 is the '
J
characteristic equation of the matrix M, but since det (M-Jl)=det(_I-_-B), then
(let ( _ _"I- /_A-B) = 0 is also the characteristic equation for M, and it is a polynomial
equation in terms of the matrices A and B which are submatrices of the original
matrix M. Assume first /_=0, then M =[_ ;] and even though wecannot use the
theorem just proved (S = r) is not nonsingular) we can still see that det M = (-l)'_.det B
22-12
]986004609-339
by Laplace'sexpansion.To compute thedeterminantof M-_ Iinthiscase we have det
i (M- _ I) =Jet 16 "El and since -_I is nensingular, ;he theorem does apply in this case toI.
.I
give us
det (M- A I)= det ()_zI-B) = (-l)']-det (B-_'I). Since for a fixed n, (-I) " is a
constant, it will not affect the eigenvalues in the polynomial characteristic equation.
t
Thus if,_x, i = I, ..., n are the eigenvalues of B, then _ = t _I/_-_ will be the eigenvalues
' of M. (Note: det (B-/f;I) = 0 implies det (M-_,.I) = 0.)
Assume now that B is diagonalizable. Then for each_(;, i = I, 2, ..., n there exists an
,. - eigenvector UI , i = I, 2, ..., n such that the set (U l , U_, ..., Un) of eigenvectors is
linearly l_,dependent. Thus let T = (U) , U_, ..., Un) be the matrix of eigenvectors as
columns in the matrix. Then T-IBT = D (_Ys,_'_, "*) _, ) will be the diagonal matrix of
eigenvalues.
Let S = _.. be the matrix where the
4q
_/'s here are the positive square roots of the_.'s and thus are half of the eigenvalues of
M. Consider first the case whereg.t0 for all i = I, 2, ..., n. Then S -I exists and .
,,..¢.,j• Inthiscaselet P :Lr __.]andP-= _L-'r-'-r-'J
. ][: ]p_, r,T,-,or MP:_Ls_"-T""ojLT -T -iL-T"ers"vs -r"_rS"-
f ,.
' .,--,_-, r%o][;,.o]=r',,,:o].r_,o],, _, Computing BT we obtain, LO ,._.j , _. Lo '"_I ""-,, LO_.J
" 22-13
! ,
1986004609-340
_m_ "_" " '_t ] ' , - ' i
F',,,ol T-,
:L 0 Jd= s. Since BT diagonalizes B to its eigenvalue matrix.
[:_s]Thus P-_ MP = ½[-S+_; -S-S --- "
l
Hence this choice for P will diag_alize M where the positive square roots of,_ are
down the diagonal of the upper-left n x n matrix aM the negative square roots of ei are
the diagonal elements of the lower right n x n matrix. Thus we have shown how to
• diagonalize M given any diagonalizable B provided the eigenvalues of B are nonzero. °
J
: Thus M is diagonalizable when B is diagonalizable provided none of B's eigenvalues are
%
.- - zerot and provided that A = 0.
_..
Now suppose A = ,8 I, where _ is any n,'nzero scalar. We then obtain
det(M -_I)= det(AzI- _(/31)-B)
" :det((4tA_ )l-B)
- = (-l)_.det (B_(_L,_ )I).
Again let _., i = 1, 2, ..., n be the eigenvalues of B. We then have _- _,,_ =_. or
_- _./3-,%.= 0, for i = 1, 2, ..., n. By solving these quadratic equations we again obtain
the eigenvalues for M from the eigenvalues for B.
If we assume _ _0 for all i and that all of the eigenvalues for B are distinct, then the
eigenvalues for M will be distinct, and thus M is diagonalizab_e.
\
• ®
1986004609-341
.f
References
: I. Bond, V. R., "Error Propagation in the Numerical Solutions of the Differential
i/: Equations of Orbital Mechanics", Celestial Mechanics, Vol. 27, No. l pp. 65-77, May
! 1982.
i
: 2. Bond, V. R. and Horn, M. K., "An Explicit Solution of the Linear Variational
Equations of the Two-Body Problem", NASA 7$-FM-47, 35C-1###0, September 1978.
%.
h_
_-. 3. Edelen, D. G. B. and gydoniels, A. D. An Introduction to Linear Algebra for _,
. Science and Engineering, Elsevier, Lectures 12, 13 and 17, 1976.
:-
z
#. Hohn, F. E., Elementary Matrix Algebra, Macmillan, Chapters 2, # and 8, 196#.
5. Murdoch, D. C, Linear Algebra for Underl_raduates, John Wiley & Sons, Chapter 7,
1957.
6. Stein, F. M., Introduction to Matrices and Determinants, Wadsworth, Chapters 6
and 7, 1969.
• 7. Szebehely,V.,"Three Theorems on ReiativeMotion",CelestialMechanics,Vol.I#:
pp.113-119,1976.
!
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1986004609-342


Transformation matrices between non-linear and linear differential equations

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