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Part I: Ch. 2




2 Linear Algebra



Carl Gustav Jacob Jacobi, relaxed Jacobi, in action


- Huge subject; excellent textbooks
- Many library subroutines exist
- But: "physical" matrices often simple in structure
- Specific algorithms that may (may!) be self-programmed
- We will concentrate on Relaxation Methods



2.1 General Remarks: Role of LA in Physics

Given $f(x)$, introduce finite differences
   $\Longrightarrow$ Vector $f \equiv (f_{k} ; \; k=1,\dots, M)$

Similarly, given $f(x,y)$ or $f(x,t)$
   $\Longrightarrow$ Matrix $F \equiv [f_{i,j}] \equiv [f(x_{i},y_{j}) ; \; i=1,\dots M;\;j=1,\dots N]$

Approximate the various differentials by differences:
   $\Longrightarrow$ Convert Partial Differential Equations (PDEs) into Systems of Linear Equations $ A \cdot x= b$

As a rule $A$ has a simple structure: sparse, diagonally dominated, positive definite, etc.

Fundamental manipulations:

  • Invert a matrix:
    $ \begin{eqnarray} A & \Longleftrightarrow & A^{-1} \end{eqnarray} $

  • Find the solution to the system of equations:
    $ \begin{eqnarray} A \cdot x = b \end{eqnarray} $

  • Find the eigenvalues $\lambda_{i}$ and the eigenvectors $ a_{i}$ of a quadratic matrix:

    $ \begin{eqnarray} \left. \begin{array}{r@{\quad=\quad}l} \left| A - \lambda_{i} I \right| & = 0 \\ ( A - \lambda_{i} I) \cdot a_{i} & = 0 \end{array} \right\} && \; \; \; i=1,\dots N \end{eqnarray} $


Sections in Chapter 2:


vesely 2005-10-10

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