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Matrix
Matrices
An m×n matrix is a rectangular array of complex or real
numbers arranged in m rows and n columns:
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)
▪ …
Applications
Linear systems of equations  
Eigenvalue problem 
Linear systems of equations
▪
m>n over determined system (data processing)
▪
m=n square case (what we will do)
▪
m<n under determined system
Linear systems in matrix notation
or Ax = b
Two cases for righthand coefficients
righthand coefficients b _{i} ≠ 0
Unique solution if the determinant det(A) ≠ 0
righthand coefficients b_{i} = 0
Unique solution if the determinant det(A) = 0
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}.
Gaussian elimination
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.
Numerical method = Gaussian elimination
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.
Step 2:
Repeating the same procedure to the last of two equations
gives
where
Step 3:
Doing back substitution we will find x_{2} and then x_{1}.
This direct method to find solutions for a system of
linear equations by the successive elimination is
known as Gaussian elimination.
Problems!
zero diagonal elements
roundoff errors
illconditioned systems
computational time
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