Diagonal and Diagonalizable Matrices
Diagonal and Diagonalizable Matrices
In[11]:=
MatrixForm[d]
MatrixForm[MatrixExp[d]]
Out[11]=
1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4
Out[12]=
E 0 0 0
2
0 E 0 0
3
0 0 E 0
4
0 0 0 E
Wonderful!
And if A is not diagonal, but diagonalizable? Remember that if
A=S.D.Inverse[S], then MatrixPower[A,n]=S.MatrixPower[D,n].Inverse[S], so
calculating the powers in the definition of Exp[tA] is not so bad:
Exp[At]=I+At+(At)^2/2!+(At)^3/3!+. . .
=I+SDtInv(S)+S(Dt)^2Inv[S]/2!+S(Dt)^3Inv[s]+. . .
=SInv[S]+SDtInv(S)+S(Dt)^2Inv[S]/2!+S(Dt)^3Inv[s]/3!+. . .
=S{I+Dt+(Dt)^2/2!+(Dt)^3+ . . .}Inv[S]
=SExp[Dt]Inv[S]
Let's check:
In[13]:=
MatrixForm[a]
MatrixForm[s.lambda.Inverse[s]]
Out[13]=
1 -1 0 -1 2 -1 0 -1 1
Out[14]=
1 -1 0 -1 2 -1 0 -1 1
In[15]:=
MatrixForm[MatrixExp[a]]
MatrixForm[s.MatrixExp[lambda].Inverse[s]]
Out[15]=
3 3 3
1 E E 1 E 1 E E
- + - + -- - - -- - - - + --
3 2 6 3 3 3 2 6
3 3 3
1 E 1 2 E 1 E
- - -- - + ---- - - --
3 3 3 3 3 3
3 3 3
1 E E 1 E 1 E E
- - - + -- - - -- - + - + --
3 2 6 3 3 3 2 6
Out[16]=
3 3 3
1 E E 1 E 1 E E
- + - + -- - - -- - - - + --
3 2 6 3 3 3 2 6
3 3 3
1 E 1 2 E 1 E
- - -- - + ---- - - --
3 3 3 3 3 3
3 3 3
1 E E 1 E 1 E E
- - - + -- - - -- - + - + --
3 2 6 3 3 3 2 6
O.K.
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