Matrix

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Chapter 3 Systems of Differential
Equations
Matrix Basic Definitions
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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 if and only if aij bij
for all values of i and j.
2. Addition
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 A
O O A A
3. Commutative
4. Associative
A B B A
(A B) C A (B C)
5. Multiplication (by a Scalar)
aA (a A)
in which the elements of aA are a aij
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Chapter 3 Systems of Differential
Equations
Matrix Multiplication, Inner Product
Matrix multiplication
if and only if
The product theorem
For two n n matrices A and B
In general, matrix multiplication is not
commutative !
commutator bracket symbol
But if A and B are each diagonal
associative
distributive
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Chapter 3 Systems of Differential
Equations
Matrix Multiplication, Inner Product
2 3 3 2 2 2
For example
Successive multiplication of row i of A with
column j of B row by column multiplication
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Chapter 3 Systems of Differential
Equations
Matrix Multiplication, Inner Product
For example
3 2 2 2 3 2
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Chapter 3 Systems of Differential
Equations
Unit Matrix, Null Matrix
The unit matrix 1 has elements dij, Kronecker
delta, and the property that 1A A1 A for all A
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 !
Exercise 3.2.6(a) if AB 0, at least one of
the matrices must have a zero determinant.
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Chapter 3 Systems of Differential
Equations
Direct product
--- The direct tensor or Kronecker product
If A is an m m matrix and B an n n matrix
The direct product
C is an mn mn matrix with elements
with
For instance, if A and B are both 2 2 matrices
The direct product is associative but not
commutative !
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Chapter 3 Systems of Differential
Equations
Diagonal Matrices
If a 3 3 square matrix A is diagonal
In any square matrix the sum of the diagonal
elements is called the trace.
1. The trace is a linear operation
2. The trace of a product of two matrices A and B
is independent of the order of
multiplication (even though AB ? BA)
3. The trace is invariant under cyclic
permutation of the matrices in a product.
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Chapter 3 Systems of Differential
Equations
Matrix Inversion
An operator that linearly transforms the
coordinate axes
Matrix A
Matrix A-1
An operator that linearly restore the original
coordinate axes
Where Cji is the jith cofactor of A.
The elements
and
For example
The cofactor matrix C
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Chapter 3 Systems of Differential
Equations
Matrix Inversion
For example
A (3)(-1-0)-(-1)(-2-0)(1)(4-1) -2
The elements of the cofactor matrix are
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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

Any matrix
symmetric
antisymmetric
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Chapter 3 Systems of Differential
Equations
Inverse Matrix, A-1
The reverse of the rotation
Transpose Matrix,
Defining a new matrix such that
holds only for orthogonal matrices !
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Chapter 3 Systems of Differential
Equations
Eigenvectors and Eigenvalues
A is a matrix, v is an eigenvector of the matrix
and ? the corresponding eigenvalue.
This only has none trivial solutions for det (A-
? I) 0. This gives rise to the secular equation
for the eigenvalues
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Chapter 3 Systems of Differential
Equations
Eigenvectors and Eigenvalues
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Chapter 3 Systems of Differential
Equations
Eigenvectors and Eigenvalues
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Chapter 3 Systems of Differential
Equations
Example 3.5.1 Eigenvalues and Eigenvectors
of a real symmetric matrix
The secular equation
? -1. ? xy 0, z 0
? -1,0,1
Normalized
? 0 ? x 0, y 0
Normalized
? 1 ? -xy 0, z 0
Normalized
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Chapter 3 Systems of Differential
Equations
Example 3.5.2 Degenerate
Eigenvalues
The secular equation
? -1. ? 2x 0, yz 0
? -1,1,1
Normalized
? 1 ? -yz 0 (r1 perpendicular to r2)
Normalized
? 1 ?
(r3 must be perpendicular to r1 and may be
made perpendicular to r2)
Normalized
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Chapter 3 Systems of Differential
Equations
Conversion of an nth order differential equation
to a system of n first-order differential
equations
Setting ,
, ,

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Chapter 3 Systems of Differential
Equations
Example Mass on a spring
assume
eigenvector
eigenvector
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Chapter 3 Systems of Differential
Equations
Homogeneous systems with constant coefficients
in components
y1y2-plane is called the phase plane
P (y1,y2) (0,0)
Critical point the point P at which dy2/dy1
becomes undetermined is called
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Chapter 3 Systems of Differential
Equations
Five Types of Critical points
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Chapter 3 Systems of Differential
Equations
Criteria for Types of Critical points
P is the sum of the eigenvalues, q the product
and ? the discriminant.
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Chapter 3 Systems of Differential
Equations
Stability Criteria for Critical points
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Chapter 3 Systems of Differential
Equations
Example Mass on a spring
p -c/m , q k/m and ? (c/m)2-4k/m
No damping c 0 p 0, q gt 0
? a center Underdamping c2 lt 4mk
p lt 0, q gt 0, ? lt 0 ? a stable and
attractive spiral point. Critical damping c2
4mk p lt 0, q gt 0, ? 0 ? a stable and
attractive node. Overdamping c2 gt 4mk
p lt 0, q gt 0, ? gt 0 ? a stable and attractive
node.
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Chapter 3 Systems of Differential
Equations
No basis of eigenvectors available. Degenerate
node
If matrix A has a double eigenvalue ?
since
If matrix A has a triple eigenvalue ?
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Chapter 3 Systems of Differential
Equations
No basis of eigenvectors available. Degenerate
node