[ < ] [ > ]   [ << ] [ Up ] [ >> ]         [Top] [Contents] [Index] [ ? ]

40. diag


[ < ] [ > ]   [ << ] [ Up ] [ >> ]         [Top] [Contents] [Index] [ ? ]

40.1 Functions and Variables for diag

Function: diag (lm)

Constructs a square matrix with the matrices of lm in the diagonal. lm is a list of matrices or scalars.

Example:

(%i1) load("diag")$
(%i2) a1:matrix([1,2,3],[0,4,5],[0,0,6])$
(%i3) a2:matrix([1,1],[1,0])$
(%i4) diag([a1,x,a2]);
                   [ 1  2  3  0  0  0 ]
                   [                  ]
                   [ 0  4  5  0  0  0 ]
                   [                  ]
                   [ 0  0  6  0  0  0 ]
(%o4)              [                  ]
                   [ 0  0  0  x  0  0 ]
                   [                  ]
                   [ 0  0  0  0  1  1 ]
                   [                  ]
                   [ 0  0  0  0  1  0 ]

To use this function write first load("diag").

Function: JF (lambda, n)

Returns the Jordan cell of order n with eigenvalue lambda.

Example:

(%i1) load("diag")$
(%i2) JF(2,5);
                    [ 2  1  0  0  0 ]
                    [               ]
                    [ 0  2  1  0  0 ]
                    [               ]
(%o2)               [ 0  0  2  1  0 ]
                    [               ]
                    [ 0  0  0  2  1 ]
                    [               ]
                    [ 0  0  0  0  2 ]
(%i3) JF(3,2);
                         [ 3  1 ]
(%o3)                    [      ]
                         [ 0  3 ]

To use this function write first load("diag").

Function: jordan (mat)

Returns the Jordan form of matrix mat, but codified in a Maxima list. To get the corresponding matrix, call function dispJordan using as argument the output of jordan.

Example:

(%i1) load("diag")$
(%i3) a:matrix([2,0,0,0,0,0,0,0],
               [1,2,0,0,0,0,0,0],
               [-4,1,2,0,0,0,0,0],
               [2,0,0,2,0,0,0,0],
               [-7,2,0,0,2,0,0,0],
               [9,0,-2,0,1,2,0,0],
               [-34,7,1,-2,-1,1,2,0],
               [145,-17,-16,3,9,-2,0,3])$
(%i34) jordan(a);
(%o4)             [[2, 3, 3, 1], [3, 1]]
(%i5) dispJordan(%);
                [ 2  1  0  0  0  0  0  0 ]
                [                        ]
                [ 0  2  1  0  0  0  0  0 ]
                [                        ]
                [ 0  0  2  0  0  0  0  0 ]
                [                        ]
                [ 0  0  0  2  1  0  0  0 ]
(%o5)           [                        ]
                [ 0  0  0  0  2  1  0  0 ]
                [                        ]
                [ 0  0  0  0  0  2  0  0 ]
                [                        ]
                [ 0  0  0  0  0  0  2  0 ]
                [                        ]
                [ 0  0  0  0  0  0  0  3 ]

To use this function write first load("diag"). See also dispJordan and minimalPoly.

Function: dispJordan (l)

Returns the Jordan matrix associated to the codification given by the Maxima list l, which is the output given by function jordan.

Example:

(%i1) load("diag")$
(%i2) b1:matrix([0,0,1,1,1],
                [0,0,0,1,1],
                [0,0,0,0,1],
                [0,0,0,0,0],
                [0,0,0,0,0])$
(%i3) jordan(b1);
(%o3)                  [[0, 3, 2]]
(%i4) dispJordan(%);
                    [ 0  1  0  0  0 ]
                    [               ]
                    [ 0  0  1  0  0 ]
                    [               ]
(%o4)               [ 0  0  0  0  0 ]
                    [               ]
                    [ 0  0  0  0  1 ]
                    [               ]
                    [ 0  0  0  0  0 ]

To use this function write first load("diag"). See also jordan and minimalPoly.

Function: minimalPoly (l)

Returns the minimal polynomial associated to the codification given by the Maxima list l, which is the output given by function jordan.

Example:

(%i1) load("diag")$
(%i2) a:matrix([2,1,2,0],
               [-2,2,1,2],
               [-2,-1,-1,1],
               [3,1,2,-1])$
(%i3) jordan(a);
(%o3)               [[- 1, 1], [1, 3]]
(%i4) minimalPoly(%);
                            3
(%o4)                (x - 1)  (x + 1)

To use this function write first load("diag"). See also jordan and dispJordan.

Function: ModeMatrix (A,l)

Returns the matrix M such that (M^^-1).A.M=J, where J is the Jordan form of A. The Maxima list l is the codified form of the Jordan form as returned by function jordan.

Example:

(%i1) load("diag")$
(%i2) a:matrix([2,1,2,0],
          [-2,2,1,2],
          [-2,-1,-1,1],
          [3,1,2,-1])$

(%i3) jordan(a);
(%o3)               [[- 1, 1], [1, 3]]
(%i4) M: ModeMatrix(a,%);
                  [  1    - 1   1   1 ]
                  [                   ]
                  [   1               ]
                  [ - -   - 1   0   0 ]
                  [   9               ]
                  [                   ]
(%o4)             [   13              ]
                  [ - --   1   - 1  0 ]
                  [   9               ]
                  [                   ]
                  [  17               ]
                  [  --   - 1   1   1 ]
                  [  9                ]
(%i5) is(  (M^^-1).a.M = dispJordan(%o3)  );
(%o5)                      true

Note that dispJordan(%o3) is the Jordan form of matrix a.

To use this function write first load("diag"). See also jordan and dispJordan.

Function: mat_function (f,mat)

Returns f(mat), where f is an analytic function and mat a matrix. This computation is based on Cauchy's integral formula, which states that if f(x) is analytic and mat = diag([JF(m1,n1), ..., JF(mk,nk)]), then f(mat) = ModeMatrix * diag([f(JF(m1,n1)), ..., f(JF(mk,nk))]) * ModeMatrix^^(-1). Note that there are about 6 or 8 other methods for this calculation. Some examples follow.

To use this function write first load(diag).

Example 1:

(%i1) load("diag")$
(%i2) b2:matrix([0,1,0], [0,0,1], [-1,-3,-3])$
(%i3) mat_function(exp,t*b2);
               2   - t
              t  %e          - t     - t
(%o3) matrix([-------- + t %e    + %e   ,
                 2
        - t     - t                           - t
 2    %e      %e        - t           - t   %e
t  (- ----- - ----- + %e   ) + t (2 %e    - -----)
        t       2                             t
               t
                         - t          - t     - t
       - t       - t   %e        2  %e      %e
 + 2 %e   , t (%e    - -----) + t  (----- - -----)
                         t            2       t
               2   - t            - t     - t
     - t      t  %e        2    %e      %e        - t
 + %e   ], [- --------, - t  (- ----- - ----- + %e   ),
                 2                t       2
                                         t
        - t     - t      2   - t
   2  %e      %e        t  %e          - t
- t  (----- - -----)], [-------- - t %e   ,
        2       t          2
        - t     - t                           - t
 2    %e      %e        - t           - t   %e
t  (- ----- - ----- + %e   ) - t (2 %e    - -----),
        t       2                             t
               t
      - t     - t                 - t
 2  %e      %e            - t   %e
t  (----- - -----) - t (%e    - -----)])
      2       t                   t
(%i4) ratsimp(%);
               [   2              - t ]
               [ (t  + 2 t + 2) %e    ]
               [ -------------------- ]
               [          2           ]
               [                      ]
               [         2   - t      ]
(%o4)  Col 1 = [        t  %e         ]
               [      - --------      ]
               [           2          ]
               [                      ]
               [     2          - t   ]
               [   (t  - 2 t) %e      ]
               [   ----------------   ]
               [          2           ]
         [      2        - t    ]
         [    (t  + t) %e       ]
         [                      ]
 Col 2 = [     2            - t ]
         [ - (t  - t - 1) %e    ]
         [                      ]
         [     2          - t   ]
         [   (t  - 3 t) %e      ]
         [        2   - t       ]
         [       t  %e          ]
         [       --------       ]
         [          2           ]
         [                      ]
         [      2          - t  ]
 Col 3 = [    (t  - 2 t) %e     ]
         [  - ----------------  ]
         [           2          ]
         [                      ]
         [   2              - t ]
         [ (t  - 4 t + 2) %e    ]
         [ -------------------- ]
         [          2           ]

Example 2:

(%i5) b1:matrix([0,0,1,1,1],
                [0,0,0,1,1],
                [0,0,0,0,1],
                [0,0,0,0,0],
                [0,0,0,0,0])$
(%i6) mat_function(exp,t*b1);
                  [              2     ]
                  [             t      ]
                  [ 1  0  t  t  -- + t ]
                  [             2      ]
                  [                    ]
(%o6)             [ 0  1  0  t    t    ]
                  [                    ]
                  [ 0  0  1  0    t    ]
                  [                    ]
                  [ 0  0  0  1    0    ]
                  [                    ]
                  [ 0  0  0  0    1    ]

(%i7) minimalPoly(jordan(b1)); 3 (%o7) x

(%i8) ident(5)+t*b1+1/2*(t^2)*b1^^2;
                  [              2     ]
                  [             t      ]
                  [ 1  0  t  t  -- + t ]
                  [             2      ]
                  [                    ]
(%o8)             [ 0  1  0  t    t    ]
                  [                    ]
                  [ 0  0  1  0    t    ]
                  [                    ]
                  [ 0  0  0  1    0    ]
                  [                    ]
                  [ 0  0  0  0    1    ]
(%i9) mat_function(exp,%i*t*b1);
             [                           2 ]
             [                          t  ]
             [ 1  0  %i t  %i t  %i t - -- ]
             [                          2  ]
             [                             ]
(%o9)        [ 0  1   0    %i t    %i t    ]
             [                             ]
             [ 0  0   1     0      %i t    ]
             [                             ]
             [ 0  0   0     1        0     ]
             [                             ]
             [ 0  0   0     0        1     ]

(%i10) mat_function(cos,t*b1)+%i*mat_function(sin,t*b1);

              [                           2 ]
              [                          t  ]
              [ 1  0  %i t  %i t  %i t - -- ]
              [                          2  ]
              [                             ]
(%o10)        [ 0  1   0    %i t    %i t    ]
              [                             ]
              [ 0  0   1     0      %i t    ]
              [                             ]
              [ 0  0   0     1        0     ]
              [                             ]
              [ 0  0   0     0        1     ]

Example 3:

(%i11) a1:matrix([2,1,0,0,0,0],
                 [-1,4,0,0,0,0],
                 [-1,1,2,1,0,0],
                 [-1,1,-1,4,0,0],
                 [-1,1,-1,1,3,0],
                 [-1,1,-1,1,1,2])$
(%i12) fpow(x):=block([k],declare(k,integer),x^k)$
(%i13) mat_function(fpow,a1);
                [  k      k - 1 ]         [      k - 1    ]
                [ 3  - k 3      ]         [   k 3         ]
                [               ]         [               ]
                [       k - 1   ]         [  k      k - 1 ]
                [  - k 3        ]         [ 3  + k 3      ]
                [               ]         [               ]
                [       k - 1   ]         [      k - 1    ]
                [  - k 3        ]         [   k 3         ]
(%o13)  Col 1 = [               ] Col 2 = [               ]
                [       k - 1   ]         [      k - 1    ]
                [  - k 3        ]         [   k 3         ]
                [               ]         [               ]
                [       k - 1   ]         [      k - 1    ]
                [  - k 3        ]         [   k 3         ]
                [               ]         [               ]
                [       k - 1   ]         [      k - 1    ]
                [  - k 3        ]         [   k 3         ]
         [       0       ]         [       0       ]
         [               ]         [               ]
         [       0       ]         [       0       ]
         [               ]         [               ]
         [  k      k - 1 ]         [      k - 1    ]
         [ 3  - k 3      ]         [   k 3         ]
         [               ]         [               ]
 Col 3 = [       k - 1   ] Col 4 = [  k      k - 1 ]
         [  - k 3        ]         [ 3  + k 3      ]
         [               ]         [               ]
         [       k - 1   ]         [      k - 1    ]
         [  - k 3        ]         [   k 3         ]
         [               ]         [               ]
         [       k - 1   ]         [      k - 1    ]
         [  - k 3        ]         [   k 3         ]
         [    0    ]
         [         ]         [ 0  ]
         [    0    ]         [    ]
         [         ]         [ 0  ]
         [    0    ]         [    ]
         [         ]         [ 0  ]
 Col 5 = [    0    ] Col 6 = [    ]
         [         ]         [ 0  ]
         [    k    ]         [    ]
         [   3     ]         [ 0  ]
         [         ]         [    ]
         [  k    k ]         [  k ]
         [ 3  - 2  ]         [ 2  ]

[ << ] [ >> ]           [Top] [Contents] [Index] [ ? ]

This document was generated by Jaime Villate on Oktober, 14 2014 using texi2html 1.76.