(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 11674, 538]*) (*NotebookOutlinePosition[ 12534, 569]*) (* CellTagsIndexPosition[ 12490, 565]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell["Operationen auf Skalare, Vektoren und Matrizen", \ "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Copyright", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Copyright 1994, Claudia Funke TU-Berlin, FB 13, Fachgebiet \[CapitalODoubleDot]konometrie und Statistik Dieses Notebook darf ausschlie\[SZ]lich als Unterrichtsmaterial und f\ \[UDoubleDot]r private Zwecke verwendet und nicht ohne Zustimmung der Autorin ver\[ADoubleDot]ndert \ werden. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ Die meisten in Mathematica verf\[UDoubleDot]gbaren mathematischen Operatoren \ sind so implementiert, da\[SZ] sie auf einen Vektor und eine Matrix angewendet \ diese elementweise auswerten. Im folgenden werden die wichtigsten dieser mathematischen Operatoren beschrieben. Dabei wird Ordnungskonformit\ \[ADoubleDot]t der betrachteten Vektoren und Matrizen vorausgesetzt. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Numerische Operatoren", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Addition/Subtraktion", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ A = B + C \ \>", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ mit\tA = (m,n) - Matrix,\t B = (m,n) - Matrix,\tC = (m,n) - Matrix \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ b = { {1,2}, {3,4} }; MatrixForm[b] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ c = { {c11,c12}, {c21,c22} }; MatrixForm[c] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ a = b + c; MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Wird ein Skalar zu einer Matrix addiert/subtrahiert, expandiert Mathematica diesen intern zu einer Matrix gleicher Gr\[ODoubleDot]\[SZ]e. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ A = B + s \ \>", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ mit\tA = (m,n) - Matrix,\t B = (m,n) - Matrix,\ts = (1,1) - Matrix\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}, Background->GrayLevel[1]], Cell[CellGroupData[{Cell["\<\ a = b + s; MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ MatrixForm[{{1 + s, 2 + s}, {3 + s, 4 + s}}] \ \>", "\<\ 1 + s 2 + s 3 + s 4 + s\ \>"], "Output", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Multiplikation", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Das normale Matrizenprodukt ist als \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["A = B * C", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ mit\tA = (m,n) - Matrix,\t B = (m,k) - Matrix,\tC = (k,n) - Matrix\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}, Background->GrayLevel[1]], Cell["und ", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ k a_ij = Sum b_ir c_ij r=1\ \>", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ definiert. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ d = { {4,3}, {2,5} }; MatrixForm[d] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Die Multiplikation zweier Matrizen kann numerisch \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ a = b . d; MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ wie auch symbolisch erfolgen. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ a = b . c; MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Die Multiplikation einer Matrix mit einem Vektor ist ebenfalls kein Problem. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ a = b . {v1, v2}; MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Bei der Multiplikation einer Matrix mit einem Skalar wird jedes Element der Matrix mit dem Skalar multipliziert. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ A = B * s \ \>", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ mit\tA = (m,n) - Matrix,\t B = (m,n) - Matrix,\ts = (1,1) - Matrix\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}, Background->GrayLevel[1]], Cell["\<\ a = s b; MatrixForm[a]\ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Das n-te Produkt einer Matrix \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ n A = Prod B_i i=1\ \>", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ wird mit \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["MatrixPowert[ Liste, n ]", "Special1", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ berechnet. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ b = { {b11,b12}, {b21,b22} }; MatrixForm[b] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ a = MatrixPower[b, 3]; MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Dimensions[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Bei der Reihenfolge der Operatoren mu\[SZ] nicht auf Ordnungskonformit\ \[ADoubleDot]t geachtet werden, da der dot-Operator . sowohl die M\[ODoubleDot]glichkeit \ der Pr\[ADoubleDot]multiplikation als auch der Postmultiplikation \ pr\[UDoubleDot]ft und die Operation ausf\[UDoubleDot]hrt, die m\[ODoubleDot]glich ist. F\[UDoubleDot]r das \ Multiplizieren von Vektoren gilt, da\[SZ] Mathematica nur Listen kennt und daher ebenfalls nicht auf Ordnungs- konformit\[ADoubleDot]t in der Reihenfolge der Operanden pr\[UDoubleDot]ft. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Spezielle Vektorprodukte", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Das innere Produkt zweier Vektoren (Skalarprodukt) \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["z = u' * v", "Special1", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ mit\tz = (1,1) - Matrix,\t u = (n,1) - Matrix,\tv = (n,1) - Matrix\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}, Background->GrayLevel[1]], Cell["und", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ n Sum u_i v_i i=1\ \>", "Special1", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ wird ebenfalls \[UDoubleDot]ber den dot-Operator berechnet. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ u = {1, 2, 3}; v = {v1, v2, v3}; \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ z = u . v \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Das \[ADoubleDot]u\[SZ]ere Produkt zweier Vektoren (dyadisches Produkt) \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Z = u * v'", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ mit\tZ = (m,n) - Matrix,\t u = (m,1) - Matrix,\tv = (n,1) - Matrix\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}, Background->GrayLevel[1]], Cell["\<\ und \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Z_ij = v_i u_j i = 1, 2, ... , n; j = 1, 2, ..., n", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ wird mit der Funktion \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Outer[ Times, Liste1, Liste2 ]", "Special1", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ berechnet. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ z = Outer[Times, u, v]; MatrixForm[z] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Kronekerprodukt", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Mit der gleichen Funktion wird auch das Kroneckerprodukt ermittelt. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["A = B \[CapitalOSlash] C ", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ mit\tA = (mp,nq) - Matrix,\t u = (m,n) - Matrix,\tv = (p,q) - Matrix \t\[CapitalOSlash] ist der Ersatz f\[UDoubleDot]r das hier nicht darstellbare \ Zeichen des \tKroneckerproduktes\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}, Background->GrayLevel[1]], Cell["mit", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["a_ij = b_ij C, i = 1, 2, ... , n; j = 1, 2, ..., n", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ b = { {b11,b12},{b21,22} }; c = { {c11,c12}, {c21,c22}, {c31,c32} }; a = Outer[Times, b, c]\ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Dimensions[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Potenzen", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Der Potenzoperator ^ kann ebenfalls auf eine Matrix angewandt werden und ergibt \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["a_ij = b_ij^n", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ b = { {b11,b12}, {b21,b22} }; a = b^2; MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Der Exponent selbst kann auch eine Matrix sein und es ergibt sich dann \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["a_ij = b_ij^c_ij", "Special2", TextAlignment->Center, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ c = { {c11,c12}, {c21,c22} }; a = b^c; MatrixForm[a] \ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{499, 599}, WindowMargins->{{Automatic, 76}, {23, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False} ] (******************************************************************* Cached data follows. 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