# 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, A-B, 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 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.

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 a21/a11 from the
second one, and multiplied by the coefficient a31/a11 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 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. Problems! zero diagonal elements round-off errors ill-conditioned systems computational time

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