(* 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[ 246711, 4413] NotebookOptionsPosition[ 243817, 4325] NotebookOutlinePosition[ 244307, 4344] CellTagsIndexPosition[ 244264, 4341] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.523967875480302*^9, 3.523967875868861*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.523967876812587*^9, 3.5239752129985*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.523967891680769*^9}], Cell[CellGroupData[{ Cell["Spectral Bounds", "Title", CellChangeTimes->{{3.5239678967778473`*^9, 3.523967902571023*^9}}], Cell["\<\ Nicholas Wheeler 2 September 2011\ \>", "Text", CellChangeTimes->{{3.523967908461011*^9, 3.523967920428646*^9}}, FontSize->10], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.523967945404701*^9}], Cell[TextData[{ "While browsing in M. 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Ziegler's ", StyleBox["Proofs from THE BOOK", FontSlant->"Italic"], " (1998) this morning\[LongDash]it has for several weeks provided my primary \ bathroom reading material\[LongDash]I came upon a theorem first obtained by \ Laguerre, and presented here as a consequence of the inequalities " }], "Text", CellChangeTimes->{{3.523968000764608*^9, 3.523968040457802*^9}, { 3.5239680950921926`*^9, 3.52396813619343*^9}, {3.5239681681814613`*^9, 3.523968238348241*^9}, {3.523968287818494*^9, 3.523968291967306*^9}, { 3.523968332438759*^9, 3.523968395444001*^9}, {3.52396842904242*^9, 3.523968450206422*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"harmonic", " ", "mean"}], " ", "\[LessSlantEqual]", " ", RowBox[{"geometric", " ", "mean"}], " ", "\[LessSlantEqual]", " ", RowBox[{"arithmetic", " ", "mean"}]}]], "DisplayFormula", CellChangeTimes->{{3.5239684604571877`*^9, 3.523968509886692*^9}}], Cell["THEOREM: Suppose all roots of the polynomial", "Text", CellChangeTimes->{{3.523968558032909*^9, 3.523968583496952*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "n"], "+", RowBox[{ SubscriptBox["a", RowBox[{"n", "-", "1"}]], SuperscriptBox["x", RowBox[{"n", "-", "1"}]]}], "+", RowBox[{ SubscriptBox["a", RowBox[{"n", "-", "2"}]], SuperscriptBox["x", RowBox[{"n", "-", "2"}]]}]}], " ", "..."}], " ", "+", SubscriptBox["a", "0"]}]], "DisplayFormula", CellChangeTimes->{{3.523968588997184*^9, 3.523968646475713*^9}, { 3.523975315734182*^9, 3.523975327521276*^9}}], Cell["\<\ are real. Then the roots are contained in the interval with endpoints\ \>", "Text", CellChangeTimes->{{3.523968670932632*^9, 3.523968699349861*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ SubscriptBox["a", RowBox[{"n", "-", "1"}]], "n"]}], "\[PlusMinus]", " ", RowBox[{ FractionBox[ RowBox[{"n", "-", "1"}], "n"], SqrtBox[ RowBox[{ SuperscriptBox[ SubscriptBox["a", RowBox[{"n", "-", "1"}]], "2"], "-", RowBox[{ FractionBox[ RowBox[{"2", "n"}], RowBox[{"n", "-", "1"}]], SubscriptBox["a", RowBox[{"n", "-", "2"}]]}]}]]}]}]], "DisplayFormula", CellChangeTimes->{{3.523968705968382*^9, 3.523968812404517*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.523968861737029*^9}], Cell["\<\ I am curious to know whether this pretty result permits one to place useful \ bounds on the spectra of finite-dimensional hermitian (or real symmetric) \ matrices.\ \>", "Text", CellChangeTimes->{{3.5239689133459187`*^9, 3.5239690007317753`*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.523969037077259*^9}], Cell["\<\ The eigenvalues of a real or complex hermitian matrix are known to be real. \ The characteristic polynomial of such a matrix \[DoubleStruckCapitalA] can be \ developed (I quote now from my \"Trace of Inverted Matrix\" [pdf file: \ December 1996])\ \>", "Text", CellChangeTimes->{{3.523969044376*^9, 3.5239691473440847`*^9}, { 3.523971331908082*^9, 3.5239714676546717`*^9}, {3.523972009920673*^9, 3.523972012272326*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "n"], RowBox[{ FractionBox["1", RowBox[{"k", "!"}]], SubscriptBox["Q", "k"], SuperscriptBox[ RowBox[{"(", RowBox[{"-", "x"}], ")"}], RowBox[{"n", "-", "k"}]]}]}], "=", RowBox[{ RowBox[{ SubscriptBox["Q", "0"], SuperscriptBox[ RowBox[{"(", RowBox[{"-", "x"}], ")"}], "n"]}], "+", RowBox[{ SubscriptBox["Q", "1"], SuperscriptBox[ RowBox[{"(", RowBox[{"-", "x"}], ")"}], RowBox[{"n", "-", "1"}]]}], "+", RowBox[{ FractionBox["1", "2"], SubscriptBox["Q", "2"], SuperscriptBox[ RowBox[{"(", RowBox[{"-", "x"}], ")"}], RowBox[{"n", "-", "2"}]]}], "+", "\[CenterEllipsis]"}]}]], "DisplayFormula", CellChangeTimes->{{3.523971553003044*^9, 3.5239717544619303`*^9}}], Cell["where", "Text", CellChangeTimes->{{3.523971874834215*^9, 3.52397187539185*^9}}], Cell[BoxData[{ RowBox[{ SubscriptBox["Q", "0"], "=", "1"}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["Q", "1"], "=", SubscriptBox["T", "1"]}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["Q", "2"], "=", RowBox[{ SuperscriptBox[ SubscriptBox["T", "1"], "2"], "-", SubscriptBox["T", "2"]}]}], "\[IndentingNewLine]", RowBox[{" ", "\[VerticalEllipsis]"}]}], "DisplayFormula", CellChangeTimes->{{3.523971886298065*^9, 3.523971961187397*^9}}], Cell[TextData[{ "with ", Cell[BoxData[ RowBox[{ SubscriptBox["T", "k"], "=", RowBox[{"Tr", "[", SuperscriptBox["\[DoubleStruckCapitalA]", "k"], "]"}]}]], CellChangeTimes->{{3.5239719904192944`*^9, 3.523972036818653*^9}}], ". To render the polynomial described above into the monic form presumed by \ the theorem, we divide by ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", "-", ")"}], "n"], SubscriptBox["Q", "0"]}], "=", SuperscriptBox[ RowBox[{"(", "-", ")"}], "n"]}]], CellChangeTimes->{{3.523972203540374*^9, 3.52397227213663*^9}}], ", to obtain" }], "Text", CellChangeTimes->{{3.523971980890893*^9, 3.5239719817498627`*^9}, { 3.523972052554554*^9, 3.523972055286057*^9}, {3.523972120212227*^9, 3.523972186206847*^9}, {3.523972295724115*^9, 3.523972307641433*^9}, 3.523972647879033*^9}], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["x", "n"], "-", RowBox[{ SubscriptBox["T", "1"], SuperscriptBox["x", RowBox[{"n", "-", "1"}]]}], "+", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox[ SubscriptBox["T", "1"], "2"], "-", SubscriptBox["T", "2"]}], "2"], SuperscriptBox["x", RowBox[{"n", "-", "2"}]]}], "+", "\[CenterEllipsis]"}], " ", "\[Congruent]", RowBox[{ SuperscriptBox["x", "n"], "+", RowBox[{ 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of by-hand repetition of the preceding command we can automatically \ do so many time, and plot the results\[Ellipsis]which will serve to expose \ the ", StyleBox["regularity", FontSlant->"Italic"], " of the results. 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