Chapter 3
Systems of Differential Equations
Matrix – Basic Definitions
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Matrix – Properties
Matrices A, B and C with elements aij, bij and cij, respectively.
1. Equality
For A and B each be m by n arrays
Matrix A = Matrix B
2. Addition
if and only if aij = bij for all values of i and j.
For A , B and C each be m by n arrays
A+B=C
if and only if aij + bij = cij for all values of i and j.
If B = O (the null matrix), for all A :
0

0
O  
..

0
0
0
..
0
..
..
..
..
A+O = O +A=A
0

0
.. 

0 
3. Commutative
A+B=B+A
5. Multiplication (by a Scalar)
αA = (α A)
4. Associative
(A + B) + C = A + (B + C)
in which the elements of αA are α aij
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Matrix Multiplication, Inner Product
Matrix multiplication
AB  C
if and only if
c ij   a ik b kj
k
The product theorem
For two n × n matrices A and B
AB  A B
* In general, matrix multiplication is not commutative !
AB  BA
commutator bracket symbol [ A, B]  AB  BA  0
But if A and B are each diagonal
* associative
(AB)C  A( BC)
* distributive
A( B  C)  AB  AC
AB  BA
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Matrix Multiplication, Inner Product
For example :
[2 × 3] × [3 × 2] = [2 × 2]
 a11 a12
a
 21 a22
b11 b12 
a13  
 c11 c12 

b21 b22   



a23 
c
c
b31 b32   21 22 
(a11  b11 )  (a12  b21 )  (a13  b31 )  c11
(a11  b12 )  (a12  b22 )  (a13  b32 )  c12
(a21  b11 )  (a22  b21 )  (a23  b31 )  c21
(a21  b12 )  (a22  b22 )  (a23  b32 )  c22
Successive multiplication of row i of A with column j of B – row by column
multiplication
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Matrix Multiplication, Inner Product
For example :
[3 × 2] × [2 × 2] = [3 × 2]
4 3 
4 * 2  3 *1 4 * 5  3 * 6 
 2 5 



AB  7 2 

7
*
2

2
*
1
7
*
5

2
*
6
 

1
6


9 0 
9 * 2  0 *1 9 * 5  0 * 6 
11 38 
 16 47 
18 45 
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Unit Matrix, Null Matrix
The unit matrix 1 has elements δij, Kronecker delta, and the property that 1A =
A1 = A for all A
1

0

1 
..

0
0
1
..
0
..
..
..
..
0

0
.. 

1
If A is an n × n matrix with determinant  0, then it has a unique inverse A-1 so
that AA -1 = A -1 A = 1.
The null matrix O has all elements being zero !
0

0
O  
..

0
0
0
..
0
..
..
..
..
0

0
.. 

0
Exercise 3.2.6(a) : if AB = 0, at least one of the matrices must have a zero
determinant.
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
--- The direct tensor or Kronecker product
Direct product
If A is an m × m matrix and B an n × n matrix
ABC
The direct product
C is an mn × mn matrix with elements
C  AijBkl
with
  n(i  1)  k
  n( j  1)  l
For instance, if A and B are both 2 × 2 matrices
a 11b11
a 11B a 12 B
a 11b 21
C AB(
)(
a 21B a 22 B
a 21b11
a 11b12
a 11b22
a 21b12
a 12 b11
a 12 b 21
a 22 b11
a 21b 21 a 21b22 a 22 b 21
c11
a 12 b12
c 21
a 12 b22
)(
c31
a 22 b12
c 41
a 22 b22
c12 c13
c 22 c 23
c32 c33
c 42 c 43
c14
c 24
)
c34
c 44
The direct product is associative but not commutative !
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Diagonal Matrices
If a 3 × 3 square matrix A is diagonal
 a 11 0

A   0 a 22
0 0

In any square matrix the sum of the
diagonal elements is called the trace.
0

0
a 33 
1. The trace is a linear operation :
trace ( A)   a ii
i
trace(A  B)  trace(A)  trace( B)
2. The trace of a product of two matrices A and B is independent of the order
of multiplication : (even though AB  BA)
trace( AB)   ( AB)ii   a ijb ji   b jia ij   ( BA) jj  trace( BA)
i
i
j
j
i
j
trace([A, B])  trace(AB)  trace( BA)  0
3. The trace is invariant under cyclic permutation of the matrices in a product.
trace(ABC)  trace( BCA)  trace(CAB)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Matrix Inversion
Matrix A
An operator that linearly transforms the coordinate axes
Matrix A-1
An operator that linearly restore the original coordinate axes
AA1  A1A  1
The elements
For example :
A  10
( A )ij  a ij
1
( 1 )
C ji

A
 1 2
A

  3 4
C ji
 ( A ) ij 
A
1
Where Cji is the jith cofactor of A.
and
A 0
The cofactor matrix C
 4 3
C

  2 1
1  4  2
A  

10  3 1 
1
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Matrix Inversion
For example :
3  1 1 
A  2 1 0 
1 2  1
|A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2
c11  (1), c12  (2), c13   (3),
The elements of the cofactor matrix are c  (1), c   ( 4), c  (7),
21
22
23
c31  (1), c32  (2), c33   (5),
 1 1  1  0.5  0.5 0.5 
C
1 
A 1 

2  4 2    1.0 2.0  1.0
 

A 2
 3  7 5    1.5 3.5  2.5
T
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Special matrices
A matrix is called symmetric if:
AT = A
A skew-symmetric (antisymmetric) matrix is one for which:
AT = -A
An orthogonal matrix is one whose transpose is also its inverse:
AT = A-1
1
~ 1
~
Any matrix A  2 [A  A]  2 [A  A]
symmetric antisymmetric
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Inverse Matrix, A-1
x'  A x
1
1
'
The reverse of the rotation
A A 1
x A x A Ax
1
x  A 1 x '
~
Transpose Matrix, A
~
a ji  a ij
A
Defining a new matrix
such that ~
a a  
ij
ik
jk
i
~
A  1 A A  A 1
~
AA  1
 ~a jia ik   jk
i
~
A  A 1
holds only for orthogonal matrices !
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Eigenvectors and Eigenvalues
Av   v
Av   v  0
A is a matrix, v is an eigenvector of the matrix and λ the corresponding eigenvalue.
 a11 a12

 a21 a22
a
 31 a32
a12
a13  v1 
 a11  

 
a22  
a23  v2   0
 a21
 a
a32
a33    v3 
 31
a13  v1   v1 
   
a23  v2     v2   0
a33  v3   v3 
This only has none trivial solutions for det (A- λ I) = 0. This gives rise to the
secular equation for the eigenvalues:
a11  
a12
a21
a31
a22  
a32
a13
a23  0
a33  
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Eigenvectors and Eigenvalues
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Eigenvectors and Eigenvalues
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Example 3.5.1
Systems of Differential Equations
Eigenvalues and Eigenvectors of a real symmetric matrix
The secular equation
 a 11 a 12

A   a 21 a 22
a a
 31 32
a 13   0 1 0 
 

a 23    1 0 0 
a 33   0 0 0 
λ = -1,0,1
0  x 
  1

 
 1   0  y   0
 0
 z 
0



 
 1
0
1  0 0
0
0 
λ = -1.  x+y = 0, z = 0

1 1
r1  ( , ,0)
2 2
Normalized
λ = 0  x = 0, y = 0

r2  (0,0,1)
Normalized
λ = 1  -x+y = 0, z = 0

1 1
r3  ( , ,0)
2 2
Normalized
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Example 3.5.2
Degenerate Eigenvalues
The secular equation
 a 11 a 12

A   a 21 a 22
a a
 31 32
a 13   1 0 0 
 

a 23    0 0 1 
a 33   0 1 0 
λ = -1,1,1
0  x 
1   0

 
  1  y   0
 0
 0
 z 
1



 
1  0
0
0
 1 0
0
1 
λ = -1.  2x = 0, y+z = 0

1 1
r1  (0, , )
2 2
λ = 1  -y+z = 0
Normalized
(r1 perpendicular to r2)

1 1
r2  (0,
, )
2 2
Normalized
λ = 1  (r3 must be perpendicular to r1 and
may be made perpendicular to r2)
  
r3  r1  r2  (1,0,0)
Normalized
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Conversion of an nth order differential equation to a system
of n first-order differential equations
y ( n )  F( t, y, y ' ,  , y ( n 1) )
Setting
y1  y, y 2  y ' , y 3  y '' ,
y1'  y 2
y '2  y 3
y  y4
'
3
……
y'n  F( t, y1 , y 2 ,  , y n )
……
y n  y ( n 1)
y'  Ay
y  xe t
y'  xe t  Axe t
Ax  x
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Example : Mass on a spring
c ' k
y  y  y0
m
m
y  y2
''
 y1'   0
 '    k
 y 2   m
'
1
1  y 

1
c    det( A  I)  k
  y2 

m
m
assume
1  0.5
0.5x1  x 2  0
m 1
 x1 
2
x   c1  1
 
 2
eigenvector
 y1 
 2  0.5 t
 1  1.5 t
 c2 
e
 y   c1  1e

 
 1.5
 2
c2
k
c
y   y1  y 2
m
m
'
2
1
c
k
2
c
   0
 
m
m
m
k  0.75
1  1.5
1.5x1  x 2  0
 x1 
 1 
x   c 2  1.5


 2
eigenvector
y1  2c1e 0.5 t  c 2e 1.5 t
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Homogeneous systems with constant coefficients
y'  Ay
 y1 ( t ) 
y( t )  

y
(
t
)
 2 
in components
y1'  a11y1  a12 y 2
y'2  a 21y1  a 22 y 2
y1y2-plane is called the phase plane
dy 2 dy 2 / dt y '2 a 21y1  a 22 y 2

 ' 
dy1 dy1 / dt y1 a11y1  a12 y 2
P : (y1,y2) = (0,0)
Critical point : the point P at which dy2/dy1 becomes undetermined is called
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Five Types of Critical points
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Criteria for Types of Critical points
det( A  I) 
p  a11  a 22
a11  
a12
a 21
a 22  
 2  (a11  a 22 )  det A  0
q  det A  a11a 22  a12a 21   p 2  4q
2  p  q  (  1 )(   2 )  2  (1   2 )  1 2
P is the sum of the eigenvalues, q the product and  the discriminant.
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Stability Criteria for Critical points
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
Example : Mass on a spring
c ' k
y  y  y0
m
m
y  y2
''
'
1
 y1'   0
 '    k
 y 2   m
k
c
y   y1  y 2
m
m
'
2
1  y 
c  1 
  y2 
m
p = -c/m , q = k/m and  = (c/m)2-4k/m
 a center
No damping c = 0
: p = 0, q > 0
Underdamping c2 < 4mk
:
p < 0, q > 0,  < 0
 a stable and attractive spiral point.
Critical damping c2 = 4mk :
p < 0, q > 0,  = 0
 a stable and attractive node.
Overdamping c2 > 4mk
p < 0, q > 0,  > 0
 a stable and attractive node.
:
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
No basis of eigenvectors available. Degenerate node
y'  Ay
If matrix A has a double eigenvalue 
y (1)  xe t
y( 2)  xte t  ue t
y( 2)'  xe t  xte t  ue t  Ay ( 2)  Axte t  Aue t
since
x  Ax
x  u  Au
(A  I)u  x
If matrix A has a triple eigenvalue 
y
( 3)
1 2 t
 xt e  ute t  ve t
2
(A  I) v  u
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3
Systems of Differential Equations
No basis of eigenvectors available. Degenerate node
 y1' 
 4 1  y1 
 '   Ay  
 y 

1
2
y

 2 
 2
4
1
det( A  I) 
 2  6  9  (  3) 2  0
1 2  
 1 1   u1 
1
(A  3I)u  
x 



 1  1 u 2 
 1
y  c1 y  c 2 y
(1)
( 2)
 u1  0
u   1
 2  
 1  3t
 1  0 3t
 c1   e  c 2 (   t    )e
 1
 1 1
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung