diff --git a/doc/src/week34/week34.do.txt b/doc/src/week34/week34.do.txt index b60bdf733..338f15bca 100644 --- a/doc/src/week34/week34.do.txt +++ b/doc/src/week34/week34.do.txt @@ -635,77 +635,6 @@ developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly he * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website URL: "http://www.netlib.org" it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available. * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from URL: "http://www.netlib.org". -!split -===== Basic Matrix Features ===== - -!bblock Matrix properties reminder -!bt -\[ - \mathbf{A} = - \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix}\qquad -\mathbf{I} = - \begin{bmatrix} 1 & 0 & 0 & 0 \\ - 0 & 1 & 0 & 0 \\ - 0 & 0 & 1 & 0 \\ - 0 & 0 & 0 & 1 - \end{bmatrix} -\] -!et - - - -The inverse of a matrix is defined by - -!bt -\[ -\mathbf{A}^{-1} \cdot \mathbf{A} = I -\] -!et - - -|----------------------------------------------------------------------| -| Relations | Name | matrix elements | -|----------------------------------------------------------------------| -| $A=A^{T}$ | symmetric | $a_{ij}=a_{ji}$ | -| $A=\left (A^{T}\right )^{-1}$ | real orthogonal | $\sum_k a_{ik}a_{jk}=\sum_k a_{ki} a_{kj}=\delta_{ij}$ | -| $A=A^*$ | real matrix | $a_{ij}=a_{ij}^*$ | -| $A=A^{\dagger}$ | hermitian | $a_{ij}=a_{ji}^*$ | -| $A=\left(A^{\dagger}\right )^{-1}$ | unitary | $\sum_k a_{ik}a_{jk}^*=\sum_k a_{ki}^* a_{kj}=\delta_{ij}$ | -|----------------------------------------------------------------------| - -!eblock - -!split -=== Some famous Matrices === - - * Diagonal if $a_{ij}=0$ for $i\ne j$ - * Upper triangular if $a_{ij}=0$ for $i>j$ - * Lower triangular if $a_{ij}=0$ for $ij+1$ - * Lower Hessenberg if $a_{ij}=0$ for $i1$ - * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i>j+p$ - * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i