Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Matrix – Basic Definitions
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Matrix – Properties
Matrices A, B and C with elements aij, bij and cij, respectively.
1. EqualityFor A and B each be m by n arrays
Matrix A = Matrix B if and only if aij = bij for all values of i and j.
2. Addition
A + B = C if and only if aij + bij = cij for all values of i and j.
For A , B and C each be m by n arrays
3. CommutativeA + B = B + A
4. Associative(A + B) + C = A + (B + C)
If B = O (the null matrix), for all A : A + O = O + A = A
0..00
........
0..00
0..00
O
5. Multiplication (by a Scalar)αA = (α A)
in which the elements of αA are α aij
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Matrix Multiplication, Inner Product
CAB if and only if k
kjikijbac
Matrix multiplication
* In general, matrix multiplication is not commutative !
BAAB commutator bracket symbol 0BAAB]B,A[
But if A and B are each diagonal BAAB
* associative )BC(AC)AB(
* distributive ACAB)CB(A
The product theorem
For two n × n matrices A and B BAAB
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Matrix Multiplication, Inner Product
2221
1211
3231
2221
1211
232221
131211
cc
cc
bb
bb
bb
aaa
aaa
22322322221221
21312321221121
12321322121211
11311321121111
)()()(
)()()(
)()()(
)()()(
cbababa
cbababa
cbababa
cbababa
Successive multiplication of row i of A with column j of B – row by column multiplication
For example :[2 × 3] × [3 × 2] = [2 × 2]
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Matrix Multiplication, Inner Product
4518
4716
3811
6*05*91*02*9
6*25*71*22*7
6*35*41*32*4
61
52
09
27
34
AB
For example :
[3 × 2] × [2 × 2] = [3 × 2]
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Unit Matrix, Null Matrix
0..00
........
0..00
0..00
O
The unit matrix 1 has elements δij, Kronecker delta, and the property that 1A = A1 = A for all A
1..00
........
0..10
0..01
1
The null matrix O has all elements being zero !
Exercise 3.2.6(a) : if AB = 0, at least one of the matrices must have a zero determinant.
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.
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Direct product --- The direct tensor or Kronecker product
If A is an m × m matrix and B an n × n matrix
The direct product CBA
C is an mn × mn matrix with elements
klijBAC k)1i(n l)1j(n with
For instance, if A and B are both 2 × 2 matrices
)
cccc
cccc
cccc
cccc
()
babababa
babababa
babababa
babababa
()BaBa
BaBa(BAC
44434241
34333231
24232221
14131211
2222212222212121
1222112212211121
2212211222112111
1212111212111111
2221
1211
The direct product is associative but not commutative !
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Diagonal Matrices
If a 3 × 3 square matrix A is diagonal
33
22
11
a00
0a0
00a
A
In any square matrix the sum of the diagonal elements is called the trace.
i
iia)A(trace
1. The trace is a linear operation : )B(trace)A(trace)BA(trace 2. The trace of a product of two matrices A and B is independent of the order of multiplication : (even though AB BA)
)BA(trace)BA(abba)AB()AB(tracej
jjj i
ijjii j
jiiji
ii
0)BA(trace)AB(trace])B,A([trace 3. The trace is invariant under cyclic permutation of the matrices in a product. )CAB(trace)BCA(trace)ABC(trace
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Matrix Inversion
Matrix A An operator that linearly transforms the coordinate axes
Matrix A-1 An operator that linearly restore the original coordinate axes
1AAAA 11
A
Ca)A( ji)1(
ijij
1 The elements Where Cji is the jith cofactor of A.
For example :
43
21A The cofactor matrix C
12
34C
10A A
C)A( ji
ij
1
13
24
10
1A 1
and 0A
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Matrix Inversion
For example :
121
012
113
A |A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2
),1(
),1(
),1(
31
21
11
c
c
cThe elements of the cofactor matrix are
),2(
),4(
),2(
32
22
12
c
c
c
),5(
),7(
),3(
33
23
13
c
c
c
5.25.35.1
0.10.20.1
5.05.05.0
573
242
111
2
1
A
CA
T
1
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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
Any matrix ]A~
A[2
1]A
~A[
2
1A
symmetric antisymmetric
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Inverse Matrix, A-1
xAx ' '1 xAx
The reverse of the rotationxAAxAx 1'1 1AA 1
Transpose Matrix, A~
Defining a new matrix such that A~
ijjiaa~
jki
ikijaa jk
iikji
aa~ 1AA~
11 AAA~
A 1AA~
holds only for orthogonal matrices !
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Eigenvectors and Eigenvalues
vvA A is a matrix, v is an eigenvector of the matrix and λ the corresponding eigenvalue.
0 vvA
0
3
2
1
3
2
1
333231
232221
131211
v
v
v
v
v
v
aaa
aaa
aaa
0
3
2
1
333231
232221
131211
v
v
v
aaa
aaa
aaa
This only has none trivial solutions for det (A- λ I) = 0. This gives rise to the secular equation for the eigenvalues:
0
333231
232221
131211
aaa
aaa
aaa
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Eigenvectors and Eigenvalues
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Eigenvectors and Eigenvalues
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Example 3.5.1 Eigenvalues and Eigenvectors of a real symmetric matrix
000
001
010
aaa
aaa
aaa
A
333231
232221
131211
The secular equation
0
00
01
01
λ = -1,0,1
0
z
y
x
00
01
01
λ = -1. x+y = 0, z = 0
)0,2
1,
2
1(r
1
Normalized
λ = 0 x = 0, y = 0
)1,0,0(r2
λ = 1 -x+y = 0, z = 0
Normalized
Normalized)0,2
1,
2
1(r
3
Chapter 3 Systems of Differential Equations
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Example 3.5.2 Degenerate Eigenvalues
010
100
001
aaa
aaa
aaa
A
333231
232221
131211
The secular equation
0
10
10
001
λ = -1,1,1
0
z
y
x
10
10
001
λ = -1. 2x = 0, y+z = 0
)2
1,
2
1,0(r
1
Normalized
λ = 1 -y+z = 0 (r1 perpendicular to r2)
)2
1,
2
1,0(r
2
λ = 1
Normalized
Normalized)0,0,1(rrr213
(r3 must be perpendicular to r1 and may be made perpendicular to r2)
Chapter 3 Systems of Differential Equations
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,,y,y,t(Fy )1n(')n(
Setting , , , ……yy1 '2 yy ''
3 yy )1n(n yy
2'1 yy
3'2 yy
4'3 yy
……
)y,,y,y,t(Fy n21'n
Ayy '
xAx
txey tt' Axexey
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3 Systems of Differential Equations
Example : Mass on a spring
0ym
ky
m
cy ''' 2
'1 yy 21
'2 y
m
cy
m
ky
2
1
'2
'1
y
y
m
c
m
k10
y
y0
m
k
m
c
m
c
m
k1
)IAdet( 2
5.01 5.11
assume 1m 2c 75.0k
0xx5.0 21 eigenvector
1
2c
x
x1
2
1
0xx5.1 21
5.1
1c
x
x2
2
1
eigenvector
t5.12
t5.01
2
1 e5.1
1ce
1
2c
y
y
t5.12
t5.011 ecec2y
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 3 Systems of Differential Equations
Homogeneous systems with constant coefficients
Ayy ' in components212111
'1 yayay
222121'2 yayay
)t(y
)t(y)t(y
2
1y1y2-plane is called the phase plane
Critical point : the point P at which dy2/dy1 becomes undetermined is called
212111
222121'1
'2
1
2
1
2
yaya
yaya
y
y
dt/dy
dt/dy
dy
dy
P : (y1,y2) = (0,0)
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
0Adet)aa(aa
aa)IAdet( 2211
2
2221
1211
2211 aap 21122211 aaaaAdetq q4p2
21212
212 )())((qp
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
0ym
ky
m
cy ''' 2
'1 yy 21
'2 y
m
cy
m
ky
2
1
'2
'1
y
y
m
c
m
k10
y
y
p = -c/m , q = k/m and = (c/m)2-4k/m
No damping c = 0 : p = 0, q > 0 a center
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
Ayy ' If matrix A has a double eigenvalue
t)1( xey tt)2( uextey
tt)2(ttt)'2( AueAxteAyuextexey
since Axx
Auux xu)IA(
If matrix A has a triple eigenvalue
ttt2)3( veuteext2
1y uv)IA(
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
2
1
'2
'1
y
y
21
14Ay
y
y0)3(96
21
14)IAdet( 22
1
1x
u
u
11
11u)I3A(
2
1
1
0
u
u
2
1
t32
t31
)2(2
)1(1 e)
1
0t
1
1(ce
1
1cycycy
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