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Diagonalisation

  Given a square matrix A, DIAG returns 7 levels:

Examples:

  1. displaymath1150

    returns:

    7: 0
    6: { { '1/0' '1/0' '1/0' }
         { '1/0' '1/0' '1/0' }
         { '1/0' '1/0' '1/0' } }
    5: { [[1 0 0]  [[-2 -1 0]  [[1 1 1]
          [0 1 0]   [0 -2 -1]   [1 1 1]
          [0 0 1]   [-1 0 -2]]  [1 1 1]] }
    4: {1 -3 3 0}
    3: {1 -3 3 0}
    2: { :0: {1 1 1}
         :<<(3,0)/2 (0,1)/2 3>>: {1 '(-1,0)/2+(0,-1)/2*V3' '(-1,0)/2+(0,1)/2*V3'}
         :<< (3,0)/2 (0,-1)/2 3>>: {1 '(-1,0)/2+(0,1)/2*V3' '(-1,0)/2+(0,-1)/2*V3'}
    1: {0 1 '(3,0)/2+(0,1)/2*V3' 1 '(3,0)/2+(0,1)/2*V3' 1 }
    This means that A has 3 eigenvalues tex2html_wrap_inline1175 , and a basis of eigenvectors is:

    displaymath1151

    corresponding to tex2html_wrap_inline1177 . The characteristic and minimal polynomial are identical (this is generically the case) X3-3X2+3X. The matrix is not invertible ('1/0' is infinite) and has a 0 determinant.

  2. For the identity matrix I2, we get:
    7: 1
    6: { { 1 0 } { 0 1 } }
    5: { [[1 0]
          [0 1]] }
    4: {1 -2 1}
    3: {1 -1}
    1: { :1, Eigen: { 0 1 } :1, Eigen: { 1 0 } }
    2: {1 2}
    The minimal polynomial is X-1, different form the characteristic polynomial (X-1)2=X2-2X+1.
  3. displaymath1152

  4. A formal example:
    { { 1 A }
      { A 1 } }
  5. In dimension greater than 2, the factorisation routines may fail. For this reason, you may call MAD, factor the characteristic polynomial (e.g. by trying the FCTR instruction of ALG48) before calling DIAG. If you have ALG48 installed, try e.g.
    { { 1 1 X }
      { 1 X 1 }
      { X 1 1 } }
    MAD FCTR DIAG
    Note that this example is solved by typing DIAG directly but it may fail in other situations.
  6. Jordan cycles example:

    displaymath1153

    returns:

    7: -4
    6: : { { '1/4' '1/4' '-1/4' }
           { '-3/4' '5/4' '-1/4' }
           { '-1/2' '1/2' '1/2'  } }
    5: { [[ 1 0 0]  [[ -2 -1  1]  [[ 1 1 -1]
          [ 0 1 0]   [  2 -5  1]   [-3 5 -1]
          [ 0 0 1]]  [  1 -1 -3]]  [-2 2  2]] }
    4: {1 -5 8 -4}
    3: {1 -5 8 -4}
    2: { :2, Char: { 2 2 1 }  :2, Eigen:{ 1 1 0 }  :1: { 0 1 1 } }
    1: { 2 2 1 1}
    This means that 2 has multiplicity 2, but the corresponding eigenspace is only 1-dimensional (spanned by (1,1,0) the last vector of the Jordan chain). The first vector (2,2,1) is only a characteristic vector, his image by (A-2I) is the eigenvector (1,1,0) .

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