(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 266165, 7503] NotebookOptionsPosition[ 254922, 7144] NotebookOutlinePosition[ 255572, 7169] CellTagsIndexPosition[ 255485, 7164] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.523463933187399*^9, 3.523463933675951*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.5234639349415293`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ { SubscriptBox["a", "1"], SubscriptBox["a", "2"]}, { SubscriptBox["b", "1"], SubscriptBox["b", "2"]} }], "\[NoBreak]", ")"}]], "Input", CellChangeTimes->{{3.523463957897644*^9, 3.5234639744924717`*^9}}], Cell[BoxData[ 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Open ]], Cell["\<\ we see that many of the elements of the inverted transition matrix cannot be \ interpreted as \"transition probabilities,\" (some fall outside the interval \ [0, 1], some even wear negative signs) though they do\[LongDash]for \ non-obvious reasons\[LongDash]conform to one of the characteristic features \ of transition matrices: the elements of each column sum to unity:\ \>", "Text", CellChangeTimes->{{3.523468212870564*^9, 3.523468371816869*^9}, { 3.5234787222815104`*^9, 3.523478724246374*^9}, {3.523546573899076*^9, 3.523546700896597*^9}, {3.523546742547714*^9, 3.523546778066777*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Total", "[", RowBox[{ RowBox[{"Transpose", "[", "\[DoubleStruckCapitalU]", "]"}], "\[LeftDoubleBracket]", "k", "\[RightDoubleBracket]"}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.523466320813504*^9, 3.523466340491846*^9}, { 3.523466404835247*^9, 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