12.4 Foundation to Solve NLSP

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Solving NLSP Using Elementary Matrices

Let $A \in R^{n \times n}$ be nonsingular and $\vec{b} \in R^{n}$ be "given." Then, to solve the 12.1 The Square Linear-Systems Problem,

A \cdot \vec{x} = \vec{b}

we use multiplication by a sequence of elementary matrices (9.8 Shear Matrix, 9.9 Dilation Matrix, 9.10 Transposition Matrix) to create "special" structure

\begin{aligned} \underset{U}{\underset{⏟}{E_{t} \cdot \dots \cdot E_{2} \cdot E_{1} \cdot A}} \vec{x} & = E_{t} \cdot \dots \cdot E_{2} \cdot E_{1} \cdot \vec{b} \\ U \vec{x} & = \vec{y} \end{aligned}

Then, we solve with the appropriate algorithm.

Remark. Unfortunately, elementary matrices can cause a lot of problems when working with nontrivial modeling problems. This method is also only able to do one-off problems. A better method would be 15 LU Factorization without Pivoting.
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