System of linear equations

tags
Applied maths

Such a system with \(m\) equations and \(n\) unknowns is often denoted \(Ax = b\) where \(A\) is a matrix \(m\times n\) and \(b\) is a vector of size \(m\).

There are multiple methods to solve such a systems with different sets of hypotheses.

System types

Square matrix with full rank

In the most simple case: a square matrix with full rank, the solution exists and is unique: \(x = A^{-1} b\)

Underdetermined systems

Such systems have a family of solutions parametrized by some constraint on the solution components.

They can be solved with the following equation: \[ x = A^+ b + (I - A^+ A)w \]

where \(A^+\) is the Moore-Penrose inverse of the matrix \(A\).

Overdetermined systems

In overdetermined systems, there is no actual solution. However, one can look for the best solution according to some criterion. If that criterion is the sum of squared residuals, this leads to ordinary least squares.


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