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Applied Maths - lecture 1
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Applied Mathematics Method I
Avid Farhoodfar
Lecture 1 Part 1
Four important matrices
(reviews of applied linear algebra)
Computational Science & Engineering
Gilbert Strang
Key points; Important Matrices
We are interested on their properties I will ask you about that?
We are interested on their meaningWhere they come from?
Why that matrix instead of some other ?
The numerical part is how to deal with themHow we solve a linear system with that coefficient
matrix?
What can we say about the solution?
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
Consider Matrix K What are its properties
K=
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric
K = KT
2- K is sparse
3- K is tridiagonal
and those diagonals are
constant
If n=100
the matrix size is nXn=10000
And the number of non-zeros is 298
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric
K = KT
2- K is sparse
3- K is tridiagonal
and those diagonals are
constant Toeplitz
If n=100
the matrix size is nXn=10000
And the number of non-zeros is 298
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric
K = KT
2- K is sparseIf n=100
the matrix size is nXn=10000
And the number of non-zeros is 298
3- K is tridiagonal
and those diagonals are
constant Toeplitz
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric
K = KT
2- K is sparseIf n=100
the matrix size is nXn=10000
And the number of non-zeros is 298
3- K is tridiagonal
and those diagonals are
constant Toeplitz
boundary
boundary
Fixed-Boundaries
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric2- K is sparse3- K is Toeplitz
4- Is K invertible?
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric2- K is sparse3- K is Toeplitz
4- K invertible
There is an inverse matrix K-1
where KK-1=II is a unit matrix
I =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric2- K is sparse3- K is Toeplitz
4- K invertible
There two ways to see if a matrix is invertible
or not invertible
First way is elimination
Where we clean up below diagonal
and make our matrix upper triangular U
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric2- K is sparse3- K is Toeplitz
4- K invertible
First way elimination
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
2 -1 0 0
0 3/2 -1 0
0 -1 2 -1
0 0 -1 2
2 -1 0 0
0 3/2 -1 0
0 0 4/3 -1
0 0 -1 2
2 -1 0 0
0 3/2 -1 0
0 0 4/3 -1
0 0 0 5/4
U =
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric2- K is sparse3- K is Toeplitz
4- K invertible
If an upper triangular matrix has a full set of
(non-zero) pivots it is invertible
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
2 -1 0 0
0 3/2 -1 0
0 -1 2 -1
0 0 -1 2
2 -1 0 0
0 3/2 -1 0
0 0 4/3 -1
0 0 -1 2
2 -1 0 0
0 3/2 -1 0
0 0 4/3 -1
0 0 0 5/4
U =
pivot
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric2- K is sparse3- K is Toeplitz
4- K invertible
Matrix K is so important we will see it over and over again.
Part of the purpose of this lecture is to get matrices names
and to see if they are invertible or not invertible.
What we can change in K to make it non-invertible?
(one of the things we can do we get C matrix)
Properties of Matrix K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
Consider Matrix K What are its properties
1- K is symmetric2- K is sparse3- K is Toeplitz
4- K invertible
2 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
C=
The C matrix is:
Properties of Matrix C
Consider Matrix C What are its properties
1- C is symmetric2- C is sparse3- C is Toeplitz
4- C is not invertible
2 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
C=
Properties of Matrix C
Consider Matrix C What are its properties
1- C is symmetric2- C is sparse3- C is Toeplitz
4- C is not invertible
2 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
C=
But why C is not invertible?
Properties of Matrix C
Consider Matrix C What are its properties
1- C is symmetric2- C is sparse3- C is Toeplitz
4- C is not invertible
2 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
C=
But why C is not invertible?
We can see it without eliminationC stands for Circulant.
Properties of Matrix C
Consider Matrix C What are its properties
1- C is symmetric2- C is sparse3- C is Toeplitz
4- C is not invertible
2 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
C=
But why C is not invertible?
We can see it without eliminationC stands for Circulant.
It is because each -1 diagonal
Which have three elements
circled around to the forth.
The same for 0.
So they are not constants
but the loop around
Thats a periodic matrix
But can you find a solution
to get to zero?
Properties of Matrix C
Consider Matrix C What are its properties
1- C is symmetric2- C is sparse3- C is Toeplitz
4- C is not invertible
2 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
C=
C is periodic
Cu= 02 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
We are looking for a u which
=
0
0
0
0
u can be
1
1
1
1
1
1
1
1
u=
But is it a solution?
Properties of Matrix C
Consider Matrix C What are its properties
1- C is symmetric2- C is sparse3- C is Toeplitz
4- C is not invertible
2 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
C=
C is periodic
Cu= 02 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
=
0
0
0
0
1
1
1
1
If C-1 exists?
Multiply both sides
by C-1C-1Cu= C-10
I
u 0=This is the only solution which is not true here
u
So C is not invertible
A physical explanations about K
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
K=
fixed
fixedsprings & masses
springs & masses are all the same Toeplitz
-1
-1Fixed ends
Here if there is no force
the system does not move
A physical explanation about C
2 -1 0 -1
-1 2 -1 0
0 -1 2 -1
-1 0 -1 2
C=
springs & masses are all the same Toeplitz
C is periodic
springs & masses
If there is no force
the system can still rotate
without compress of spring
No solution
because
Properties of Matrix L(T)
2 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 1
L=
Next Matrix is L What are its propertiesfixed
1- L and T are symmetric
2- L and T are sparse
3- L and T are not Toeplitz
are L and T invertible?
free
Lower end is free
L fixed-free boundary condition
T free-fixed boundary condition
free
fixed
L and T are tridiagonal
but those diagonals
are not constant
T=
fixed
free
Top end is free
1 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 2
Properties of Matrix B
1 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 1
B=
Last Matrix is B What are its properties
1- B is symmetric
2- B is sparse
is B invertible?free
Both ends are free
B free (open) boundary condition
free
free free
3- B is not Toeplitz
B is tridiagonal
but those diagonals
are not constant
Q : Is B invertible?