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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
Numerical method = Gaussian elimination
Gaussian elimination for n =3
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