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")
.
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")
.
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
.
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
.
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
.
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
.
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 ]