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1
Matrices
Matrices
An m×n matrix is a rectangular array of complex or real
numbers arranged in
m
rows and
n
columns:
mn
m
m
m
n
n
n
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
K
K
K
K
K
K
K
K
K
3
2
1
3
33
32
31
2
23
22
21
1
13
12
11
2
Types, Operations, etc.
Types: square, symmetric, diagonal, Hermithean, …
Basic operations: A+B, AB, AB (AB
≠
BA).
Square matrices
Determinant: det(A)
Inverse matrix A
1
: AA
1
= I (I is a unit matrix)
…
3
Applications
Linear systems of equations
Eigenvalue problem
4
Linear systems of equations
m>n
over determined system (data processing)
m=n
square case (what we will do)
m<n
under determined system
5
Linear systems in matrix notation
or
Ax
=
b
6
Two cases for righthand coefficients
Ö
righthand coefficients b
i
≠
0
Unique solution if the determinant det(A)
≠
0
Ö
righthand coefficients b
i
=0
Unique solution if the determinant det(A) = 0
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Analytic solutions for n=2
a
11
x
1
+ a
12
x
2
=b
1
a
21
x
1
+ a
22
x
2
=b
2
expressing the first unknown x
1
from the first equation
x
1
= (b
1
a
12
x
2
)/a
11
and substituting to the second equation we have a
single equation with one unknown x
2
.
8
Gaussian elimination
Ö
Since there is no such an operator as elimination
neither in C++ nor Fortran we should translate this
procedure to an appropriate numerical method for
solving systems of linear equations.
Ö
Numerical method = Gaussian elimination
9
Gaussian elimination for n=3
Let subtract the first equation multiplied by the coefficient a
21
/a
11
from the
second one, and multiplied by the coefficient a
31
/a
11
from the third equation.
10
Step 2:
Step 2:
Repeating the same procedure to the last of two equations
gives
where
11
Step 3:
Step 3:
Doing back substitution we will find x
2
and then x
1
.
This direct method to find solutions for a system of
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This note was uploaded on 09/29/2009 for the course PHYSICS 811 taught by Professor Godunov during the Fall '09 term at Old Dominion.
 Fall '09
 Godunov

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