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, A-B, AB (AB≠BA).
▪ Square matrices
▪ Determinant: det(A)
▪ Inverse matrix A-1: AA-1 = I (I is a unit matrix)
|Linear systems of equations|
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 right-hand coefficients
right-hand coefficients b i ≠ 0
Unique solution if the determinant det(A) ≠ 0
right-hand coefficients bi = 0
Unique solution if the determinant det(A) = 0
Analytic solutions for n=2
a11x1 + a12x2=b1
a21x1 + a22x2=b2
expressing the first unknown x 1 from the first equation
x1 = (b1 - a12x2)/a11
and substituting to the second equation we have a
single equation with one unknown x2.
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 a21/a11 from the
second one, and multiplied by the coefficient a31/a11 from the third equation.
Repeating the same procedure to the last of two equations
Doing back substitution we will find x2 and then x1.
This direct method to find solutions for a system of
linear equations by the successive elimination is
known as Gaussian elimination.
zero diagonal elements