DIAG
returns 7 levels:
where M denotes the minimal polynomial of A
Examples:
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 , and a basis of eigenvectors is:
corresponding to .
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.
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.
{ { 1 A } { A 1 } }
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 DIAGNote that this example is solved by typing
DIAG
directly but it may
fail in other situations.
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) .