Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
1997
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 1997

[Date Index] [Thread Index] [Author Index]

Search the Archive

Bug in Minor[ ]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg6212] Bug in Minor[ ]
  • From: Mario Sancho Graca <M.S.Graca at Warwick.ac.uk>
  • Date: Thu, 27 Feb 1997 02:55:16 -0500
  • Organization: University of Warwick
  • Sender: owner-wri-mathgroup at wolfram.com

The output of Minor[A,k] computes the minors of A based on submatrices
of size k*k, but it seems that the ordering of the results is wrong.
Attached you will find a short notebook illustrating the output and
comparing with the basic formal definition of minors of a matrix.
-- 
Mario Sancho Graca
University of Warwick
Department of Engineering
Coventry CV4 7AL
United Kingdom

Email: M.S.Graca at Warwick.ac.uk


(***********************************************************************

                    Mathematica-Compatible Notebook

This notebook can be used on any computer system with Mathematica 3.0,
MathReader 3.0, or any compatible application. The data for the notebook 
starts with the line of 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).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the 
word CacheID, otherwise Mathematica-compatible applications may try to 
use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info at 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[     15303,        472]*)
(*NotebookOutlinePosition[     16232,        506]*)
(*  CellTagsIndexPosition[     16160,        500]*)
(*WindowFrame->Normal*)



Notebook[{

Cell[CellGroupData[{
Cell[TextData[{
  "Bug in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " function Minor[ ]"
}], "Title"],

Cell[TextData[
"M.S. Gra\[CCedilla]a\n\nUniversity of Warwick\nDepartment of Engineering\n\
Coventry CV4 7AL\nUnited Kingdom\n\nEmail: M.S.Graca at Warwick.ac.uk"], "Author"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s function Minor[ ] computes the minors of a matrix \[ScriptCapitalA] \
correctly, but the ordering of the resultant elements is incorrect. This note \
illustrates the point."
}], "Abstract"],

Cell[CellGroupData[{

Cell[TextData["Expansion of det \[ScriptCapitalA] in cofactors"], "Subsection"],

Cell[TextData[{
  "Consider matrix \[ScriptCapitalA], ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          RowBox[{"n", "\[Cross]", "n"}], "   ", 
          RowBox[{"(", 
            RowBox[{"n", "\[GreaterEqual]", "2"}], ")"}]}], 
        TraditionalForm]]],
  ". The determinant of \[ScriptCapitalA] is a combination of row ",
  StyleBox["i",
    FontSlant->"Italic"],
  " and the cofactors of row ",
  StyleBox["i",
    FontSlant->"Italic"],
  ":"
}], "Text"],

Cell[TextData[Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{"det", " ", "\[ScriptCapitalA]"}], "=", 
        RowBox[{
          RowBox[{
            SubscriptBox["a", "i1"], 
            SubscriptBox["\[ScriptCapitalA]", "i1"]}], "+", 
          RowBox[{
            SubscriptBox["a", "i2"], 
            SubscriptBox["\[ScriptCapitalA]", "i2"]}], "+", "\[Ellipsis]", 
          "+", 
          RowBox[{
            SubscriptBox["a", 
              RowBox[{"i", "\[VeryThinSpace]", "n"}]], 
            SubscriptBox["\[ScriptCapitalA]", 
              RowBox[{"i", "\[VeryThinSpace]", "n"}]]}]}]}], 
      TraditionalForm]]]], "Text"],

Cell[TextData[{
  StyleBox["Definition.",
    FontWeight->"Bold"],
  " The ",
  StyleBox["minor",
    FontWeight->"Bold"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\[ScriptCapitalM]", 
          RowBox[{"i", "\[VeryThinSpace]", "j"}]], TraditionalForm]]],
  " is the determinant of the ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          RowBox[{"(", 
            RowBox[{"n", "-", "1"}], ")"}], 
          RowBox[{"(", 
            RowBox[{"n", "-", "1"}], ")"}]}], TraditionalForm]]],
  " matrix formed by deleting row ",
  StyleBox["i",
    FontSlant->"Italic"],
  " and column ",
  StyleBox["j",
    FontSlant->"Italic"],
  " of \[ScriptCapitalA]."
}], "Text"],

Cell[TextData[{
  StyleBox["Definition.",
    FontWeight->"Bold"],
  " The ",
  StyleBox["cofactor",
    FontWeight->"Bold"],
  " ",
  Cell[BoxData[
      FormBox[
        SubscriptBox["\[ScriptCapitalC]", 
          RowBox[{"i", "\[VeryThinSpace]", "j"}]], TraditionalForm]]],
  " is the minor with corrected sign,"
}], "Text"],

Cell[TextData[Cell[BoxData[
    FormBox[
      RowBox[{
        SubscriptBox["\[ScriptCapitalC]", 
          RowBox[{"i", "\[VeryThinSpace]", "j"}]], "=", 
        RowBox[{
          SuperscriptBox[
            RowBox[{"(", 
              RowBox[{"-", "1"}], ")"}], 
            RowBox[{"i", "+", "j"}]], 
          SubscriptBox["\[ScriptCapitalM]", 
            RowBox[{"i", "\[VeryThinSpace]", "j"}]]}]}], TraditionalForm]]]], 
  "Text"],

Cell[CellGroupData[{

Cell[TextData["\[ScriptCapitalA] = Array[a, {3, 3}]"], "Input",
  CellTags->"S3.7.7"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{
        RowBox[{"{", 
          RowBox[{
            RowBox[{"a", "[", 
              RowBox[{"1", ",", "1"}], "]"}], ",", 
            RowBox[{"a", "[", 
              RowBox[{"1", ",", "2"}], "]"}], ",", 
            RowBox[{"a", "[", 
              RowBox[{"1", ",", "3"}], "]"}]}], "}"}], ",", 
        RowBox[{"{", 
          RowBox[{
            RowBox[{"a", "[", 
              RowBox[{"2", ",", "1"}], "]"}], ",", 
            RowBox[{"a", "[", 
              RowBox[{"2", ",", "2"}], "]"}], ",", 
            RowBox[{"a", "[", 
              RowBox[{"2", ",", "3"}], "]"}]}], "}"}], ",", 
        RowBox[{"{", 
          RowBox[{
            RowBox[{"a", "[", 
              RowBox[{"3", ",", "1"}], "]"}], ",", 
            RowBox[{"a", "[", 
              RowBox[{"3", ",", "2"}], "]"}], ",", 
            RowBox[{"a", "[", 
              RowBox[{"3", ",", "3"}], "]"}]}], "}"}]}], "}"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    RowBox[{"\[ScriptCapitalA]", "//", "MatrixForm"}]], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", GridBox[{
            {
              RowBox[{"a", "[", 
                RowBox[{"1", ",", "1"}], "]"}], 
              RowBox[{"a", "[", 
                RowBox[{"1", ",", "2"}], "]"}], 
              RowBox[{"a", "[", 
                RowBox[{"1", ",", "3"}], "]"}]},
            {
              RowBox[{"a", "[", 
                RowBox[{"2", ",", "1"}], "]"}], 
              RowBox[{"a", "[", 
                RowBox[{"2", ",", "2"}], "]"}], 
              RowBox[{"a", "[", 
                RowBox[{"2", ",", "3"}], "]"}]},
            {
              RowBox[{"a", "[", 
                RowBox[{"3", ",", "1"}], "]"}], 
              RowBox[{"a", "[", 
                RowBox[{"3", ",", "2"}], "]"}], 
              RowBox[{"a", "[", 
                RowBox[{"3", ",", "3"}], "]"}]}
            }], ")"}],
      (MatrixForm[ #]&)]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Det[ \[ScriptCapitalA] ]"], "Input",
  CellTags->"S3.7.7"],

Cell[BoxData[
    RowBox[{
      RowBox[{
        RowBox[{"-", 
          RowBox[{"a", "[", 
            RowBox[{"1", ",", "3"}], "]"}]}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"2", ",", "2"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"3", ",", "1"}], "]"}]}], "+", 
      RowBox[{
        RowBox[{"a", "[", 
          RowBox[{"1", ",", "2"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"2", ",", "3"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"3", ",", "1"}], "]"}]}], "+", 
      RowBox[{
        RowBox[{"a", "[", 
          RowBox[{"1", ",", "3"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"2", ",", "1"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"3", ",", "2"}], "]"}]}], "-", 
      RowBox[{
        RowBox[{"a", "[", 
          RowBox[{"1", ",", "1"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"2", ",", "3"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"3", ",", "2"}], "]"}]}], "-", 
      RowBox[{
        RowBox[{"a", "[", 
          RowBox[{"1", ",", "2"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"2", ",", "1"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"3", ",", "3"}], "]"}]}], "+", 
      RowBox[{
        RowBox[{"a", "[", 
          RowBox[{"1", ",", "1"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"2", ",", "2"}], "]"}], " ", 
        RowBox[{"a", "[", 
          RowBox[{"3", ",", "3"}], "]"}]}]}]], "Output"]
}, Open  ]],

Cell["The computed determinant is correct.", "Text"],

Cell[CellGroupData[{

Cell[TextData["Minors[\[ScriptCapitalA], 2]"], "Input",
  CellTags->"S3.7.7"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{
        RowBox[{"{", 
          RowBox[{
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"1", ",", "2"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "1"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"1", ",", "1"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "2"}], "]"}]}]}], ",", 
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"1", ",", "3"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "1"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"1", ",", "1"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "3"}], "]"}]}]}], ",", 
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"1", ",", "3"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "2"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"1", ",", "2"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "3"}], "]"}]}]}]}], "}"}], ",", 
        RowBox[{"{", 
          RowBox[{
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"1", ",", "2"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "1"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"1", ",", "1"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "2"}], "]"}]}]}], ",", 
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"1", ",", "3"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "1"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"1", ",", "1"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "3"}], "]"}]}]}], ",", 
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"1", ",", "3"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "2"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"1", ",", "2"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "3"}], "]"}]}]}]}], "}"}], ",", 
        RowBox[{"{", 
          RowBox[{
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"2", ",", "2"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "1"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "1"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "2"}], "]"}]}]}], ",", 
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"2", ",", "3"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "1"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "1"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "3"}], "]"}]}]}], ",", 
            RowBox[{
              RowBox[{
                RowBox[{"-", 
                  RowBox[{"a", "[", 
                    RowBox[{"2", ",", "3"}], "]"}]}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "2"}], "]"}]}], "+", 
              RowBox[{
                RowBox[{"a", "[", 
                  RowBox[{"2", ",", "2"}], "]"}], " ", 
                RowBox[{"a", "[", 
                  RowBox[{"3", ",", "3"}], "]"}]}]}]}], "}"}]}], "}"}]], 
  "Output"]
}, Open  ]],

Cell[TextData[{
  "The function Minor[ ] is computing the minors ",
  StyleBox["but",
    FontWeight->"Bold"],
  " the ordering is,\n",
  Cell[BoxData[{
      FormBox[
        RowBox[{
          RowBox[{
            RowBox[{"(", 
              RowBox[{"Minor", "[", "\[ScriptCapitalA]", "]"}], ")"}], 
            "\[LeftDoubleBracket]", 
            RowBox[{"1", ",", "1"}], "\[RightDoubleBracket]"}], "=", 
          SubscriptBox["M", 
            RowBox[{"3", ",", "3"}]]}], TraditionalForm], 
      FormBox[
        RowBox[{
          RowBox[{
            RowBox[{"(", 
              RowBox[{"Minor", "[", "\[ScriptCapitalA]", "]"}], ")"}], 
            "\[LeftDoubleBracket]", 
            RowBox[{"1", ",", "2"}], "\[RightDoubleBracket]"}], "=", 
          SubscriptBox["M", 
            RowBox[{"3", ",", "2"}]]}], TraditionalForm], 
      FormBox[
        RowBox[{"\t\t\t", 
          RowBox[{
            RowBox[{
            "\[VerticalEllipsis]", "                                      ", 
              RowBox[{
                RowBox[{"\[VerticalEllipsis]", "\n", "(", 
                  RowBox[{"Minor", "[", "\[ScriptCapitalA]", "]"}], ")"}], 
                "\[LeftDoubleBracket]", 
                RowBox[{"i", ",", "j"}], "\[RightDoubleBracket]"}]}], "=", 
            SubscriptBox["M", 
              RowBox[{
                RowBox[{"n", "-", "i", "+", "1"}], ",", 
                RowBox[{"n", "-", "j", "+", "1"}]}]]}]}], TraditionalForm]}]]
}], "Text"],

Cell[TextData[StyleBox["References",
  FontWeight->"Bold"]], "Text"],

Cell[TextData[{
  "G. Strang, 1988, ",
  StyleBox["Linear Algebra and Its Applications",
    FontSlant->"Italic"],
  ", Harcourt Brace Jovanovich Publ., Third Edition, p. 226"
}], "Reference"],

Cell[TextData[{
  "P.V. O'Neil, 1991, ",
  StyleBox["Advanced Engineering Mathematics",
    FontSlant->"Italic"],
  ", Wadsworth Publ., Third Edition, pp 688\[Dash]690"
}], "Reference"]
}, Open  ]]
}, Open  ]]
},
FrontEndVersion->"Microsoft Windows 3.0",
ScreenRectangle->{{0, 1024}, {0, 675}},
WindowSize->{761, 547},
WindowMargins->{{80, Automatic}, {Automatic, 0}},
StyleDefinitions -> "Report.nb"
]


(***********************************************************************
Cached data follows.  If you edit this Notebook file directly, not using
Mathematica, you must remove the line containing CacheID at the top of 
the file.  The cache data will then be recreated when you save this file 
from within Mathematica.
***********************************************************************)

(*CellTagsOutline
CellTagsIndex->{
  "S3.7.7"->{
    Cell[5006, 169, 85, 1, 56, "Input",
      CellTags->"S3.7.7"],
    Cell[7099, 236, 73, 1, 56, "Input",
      CellTags->"S3.7.7"],
    Cell[8780, 290, 77, 1, 56, "Input",
      CellTags->"S3.7.7"]}
  }
*)

(*CellTagsIndex
CellTagsIndex->{
  {"S3.7.7", 15933, 489}
  }
*)

(*NotebookFileOutline
Notebook[{

Cell[CellGroupData[{
Cell[1731, 51, 116, 5, 161, "Title"],
Cell[1850, 58, 171, 2, 267, "Author"],
Cell[2024, 62, 263, 6, 105, "Abstract"],

Cell[CellGroupData[{
Cell[2312, 72, 79, 0, 59, "Subsection"],
Cell[2394, 74, 470, 16, 62, "Text"],
Cell[2867, 92, 646, 17, 38, "Text"],
Cell[3516, 111, 692, 26, 64, "Text"],
Cell[4211, 139, 328, 12, 42, "Text"],
Cell[4542, 153, 439, 12, 41, "Text"],

Cell[CellGroupData[{
Cell[5006, 169, 85, 1, 56, "Input",
  CellTags->"S3.7.7"],
Cell[5094, 172, 957, 26, 116, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[6088, 203, 78, 1, 57, "Input"],
Cell[6169, 206, 893, 25, 129, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[7099, 236, 73, 1, 56, "Input",
  CellTags->"S3.7.7"],
Cell[7175, 239, 1513, 44, 141, "Output"]
}, Open  ]],
Cell[8703, 286, 52, 0, 40, "Text"],

Cell[CellGroupData[{
Cell[8780, 290, 77, 1, 56, "Input",
  CellTags->"S3.7.7"],
Cell[8860, 293, 4470, 117, 291, "Output"]
}, Open  ]],
Cell[13345, 413, 1476, 38, 142, "Text"],
Cell[14824, 453, 68, 1, 40, "Text"],
Cell[14895, 456, 192, 5, 58, "Reference"],
Cell[15090, 463, 185, 5, 58, "Reference"]
}, Open  ]]
}, Open  ]]
}
]
*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)




  • Prev by Date: Re: "Null" placed after each axis lable, what did I do wrong?
  • Next by Date: NonLinear ODE BVP
  • Previous by thread: Re: tubes
  • Next by thread: NonLinear ODE BVP