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48. diag


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48.1 Functions and Variables for diag

Function: diag (lm)

Constructs a matrix that is the block sum of the elements of lm. The elements of lm are assumed to be matrices; if an element is scalar, it treated as a 1 by 1 matrix.

The resulting matrix will be square if each of the elements of lm is square.

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 ]
(%i5) diag ([matrix([1,2]), 3]);
                        [ 1  2  0 ]
(%o5)                   [         ]
                        [ 0  0  3 ]

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

Categories:  Matrices · Share packages · Package 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").

Categories:  Package diag

Function: jordan (mat)

Returns the Jordan form of matrix mat, encoded as a list in a particular format. To get the corresponding matrix, call the function dispJordan using the output of jordan as the argument.

The elements of the returned list are themselves lists. The first element of each is an eigenvalue of mat. The remaining elements are positive integers which are the lengths of the Jordan blocks for this eigenvalue. These integers are listed in decreasing order. Eigenvalues are not repeated.

The functions dispJordan, minimalPoly and ModeMatrix expect the output of a call to jordan as an argument. If you construct this argument by hand, rather than by calling jordan, you must ensure that each eigenvalue only appears once and that the block sizes are listed in decreasing order, otherwise the functions might give incorrect answers.

Example:

(%i1) load("diag")$
(%i2) 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])$
(%i3) jordan (A);
(%o3)                [[2, 3, 3, 1], [3, 1]]
(%i4) 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 ]
(%o4)              [                        ]
                   [ 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.

Categories:  Package diag

Function: dispJordan (l)

Returns a matrix in Jordan canonical form (JCF) corresponding to the list of eigenvalues and multiplicities given by l. This list should be in the format given by the jordan function. See jordan for details of this format.

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.

Categories:  Package diag

Function: minimalPoly (l)

Returns the minimal polynomial of the matrix whose Jordan form is described by the list l. This list should be in the format given by the jordan function. See jordan for details of this format.

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.

Categories:  Package diag

Function: ModeMatrix (A, [jordan_info])

Returns an invertible matrix M such that (M^^-1).A.M is the Jordan form of A.

To calculate this, Maxima must find the Jordan form of A, which might be quite computationally expensive. If that has already been calculated by a previous call to jordan, pass it as a second argument, jordan_info. See jordan for details of the required format.

Example:

(%i1) load("diag")$
(%i2) A: matrix([2,1,2,0], [-2,2,1,2], [-2,-1,-1,1], [3,1,2,-1])$
(%i3) M: ModeMatrix (A);
                      [  1    - 1   1   1 ]
                      [                   ]
                      [   1               ]
                      [ - -   - 1   0   0 ]
                      [   9               ]
                      [                   ]
(%o3)                 [   13              ]
                      [ - --   1   - 1  0 ]
                      [   9               ]
                      [                   ]
                      [  17               ]
                      [  --   - 1   1   1 ]
                      [  9                ]
(%i4) is ((M^^-1) . A . M = dispJordan (jordan (A)));
(%o4)                         true

Note that, in this example, the Jordan form of A is computed twice. To avoid this, we could have stored the output of jordan(A) in a variable and passed that to both ModeMatrix and dispJordan.

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

Categories:  Package diag

Function: mat_function (f,A)

Returns f(A), where f is an analytic function and A a matrix. This computation is based on the Taylor expansion of f. It is not efficient for numerical evaluation, but can give symbolic answers for small matrices.

Example 1:

The exponential of a matrix. We only give the first row of the answer, since the output is rather large.

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

Example 2:

Comparison with the Taylor series for the exponential and also comparing exp(%i*A) with sine and cosine.

(%i1) load("diag")$
(%i2) A: matrix ([0,1,1,1],
                 [0,0,0,1],
                 [0,0,0,1],
                 [0,0,0,0])$
(%i3) ratsimp (mat_function (exp, t*A));
                       [           2     ]
                       [ 1  t  t  t  + t ]
                       [                 ]
(%o3)                  [ 0  1  0    t    ]
                       [                 ]
                       [ 0  0  1    t    ]
                       [                 ]
                       [ 0  0  0    1    ]
(%i4) minimalPoly (jordan (A));
                                3
(%o4)                          x
(%i5) ratsimp (ident(4) + t*A + 1/2*(t^2)*A^^2);
                       [           2     ]
                       [ 1  t  t  t  + t ]
                       [                 ]
(%o5)                  [ 0  1  0    t    ]
                       [                 ]
                       [ 0  0  1    t    ]
                       [                 ]
                       [ 0  0  0    1    ]
(%i6) ratsimp (mat_function (exp, %i*t*A));
                  [                        2 ]
                  [ 1  %i t  %i t  %i t - t  ]
                  [                          ]
(%o6)             [ 0   1     0      %i t    ]
                  [                          ]
                  [ 0   0     1      %i t    ]
                  [                          ]
                  [ 0   0     0        1     ]
(%i7) ratsimp (mat_function (cos, t*A) + %i*mat_function (sin, t*A));
                  [                        2 ]
                  [ 1  %i t  %i t  %i t - t  ]
                  [                          ]
(%o7)             [ 0   1     0      %i t    ]
                  [                          ]
                  [ 0   0     1      %i t    ]
                  [                          ]
                  [ 0   0     0        1     ]

Example 3:

Power operations.

(%i1) load("diag")$
(%i2) A: matrix([1,2,0], [0,1,0], [1,0,1])$
(%i3) integer_pow(x) := block ([k], declare (k, integer), x^k)$
(%i4) mat_function (integer_pow, A);
                       [ 1     2 k     0 ]
                       [                 ]
(%o4)                  [ 0      1      0 ]
                       [                 ]
                       [ k  (k - 1) k  1 ]
(%i5) A^^20;
                         [ 1   40   0 ]
                         [            ]
(%o5)                    [ 0    1   0 ]
                         [            ]
                         [ 20  380  1 ]

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

Categories:  Package diag


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