MatrixExp
Usage Message: MatrixExp[mat] gives the matrix exponential of mat. Attributes[MatrixExp] = {Protected}
Notes: There is an error in the initial releases of Version 3.0 for Mathematica that will cause MatrixExp to return incorrect results for certain diagonal matrices. This error does not affect any matrices other than diagonal matrices. Here is a typical example, showing an incorrect result.
In[1]:= m = DiagonalMatrix[{x, -x, x, -x}]
Out[1]= {{x, 0, 0, 0}, {0, -x, 0, 0}, {0, 0, x, 0}, {0, 0, 0, -x}}
In[2]:= MatrixExp[m]
-x -x x x
Out[2]= {{E , 0, 0, 0}, {0, E , 0, 0}, {0, 0, E , 0}, {0, 0, 0, E }}
One simple way to correct this problem is to add a rule that simply computes the exponential of the diagonal elements if MatrixExp is used with diagonal matrices.
In[3]:= Unprotect[MatrixExp]
Out[3]= {MatrixExp}
In[4]:= MatrixExp[m_List] := With[{d = Dimensions[m]},
(Print["Using diagonal MatrixExp rule"];
DiagonalMatrix[Exp[Table[m[[i,i]],{i,d[[1]]}]]]) /;
(Print["Checking diagonal MatrixExp rule"]; True) &&
Length[d] === 2 && (SameQ @@ d) &&
Catch[Do[If[i=!=j && !TrueQ[m[[i,j]]==0],
Throw[False]],
{i,d[[1]]},{j, d[[2]]}]; True]]
In[5]:= Protect[MatrixExp]
Out[5]= {MatrixExp}
In[6]:= MatrixExp[m]
Checking diagonal MatrixExp rule
Using diagonal MatrixExp rule
x -x x -x
Out[6]= {{E , 0, 0, 0}, {0, E , 0, 0}, {0, 0, E , 0}, {0, 0, 0, E }}
If you use this rule, you may want to remove the Print
statements after verifying that this rule has been installed correctly.
You can have this workaround automatically loaded by placing
http://support.wolfram.com/mathematica/kernel/Symbols/System/MatrixExp.m
at the location given by
In[7]:= ToFileName[{$TopDirectory,"AddOns", "Autoload",
"MatrixExp", "Kernel"}, "init.m"]
We apologize for any difficulties caused by this error. For more information, contact the technical support group at Wolfram Research.
There is an
error in Version 3.0 of Mathematica that can cause MatrixExp to return unnecessarily complicated results involving RootSum
expressions. Here is a typical example.
In[1]:= result = MatrixExp[{{1,1,0},{-1,2,0},{0,0,1}}]
2 3
Out[1]= RootSum[-3 + 6 #1 - 4 #1 + #1 & ,
#1 #1 #1 2 #1 #1
2 E - 3 E #1 + E #1 -E + E #1
> {{--------------------------, ----------------, 0},
2 2
6 - 8 #1 + 3 #1 6 - 8 #1 + 3 #1
#1 #1 #1 #1 #1 2
-E + E #1 E - 2 E #1 + E #1
> {-(----------------), ------------------------, 0},
2 2
6 - 8 #1 + 3 #1 6 - 8 #1 + 3 #1
#1 #1 #1 2
3 E - 3 E #1 + E #1
> {0, 0, --------------------------}} & ]
2
6 - 8 #1 + 3 #1
The result is mathematically correct, but it is possible to give this result in a simpler form, without the use of RootSum
. In most examples you can convert the result to a simpler form by using functions such as ToRadicals
, Simplify
, or Together
.
In[2]:= Together[ToRadicals[result]]
3/2 3/2 3/2 + I Sqrt[3]
Out[2]= {{(2 (-3 I E - Sqrt[3] E - 3 I E +
3/2 + I Sqrt[3]
> Sqrt[3] E )) /
I/2 Sqrt[3]
> ((-3 - I Sqrt[3]) (3 I + Sqrt[3]) E ),
3/2 3/2 + I Sqrt[3]
4 Sqrt[3] (-E + E )
> ----------------------------------------------, 0},
I/2 Sqrt[3]
(-3 + I Sqrt[3]) (-3 I + Sqrt[3]) E
3/2 3/2 + I Sqrt[3]
4 Sqrt[3] (-E + E )
> {---------------------------------------------,
I/2 Sqrt[3]
(-3 - I Sqrt[3]) (3 I + Sqrt[3]) E
3/2 3/2 3/2 + I Sqrt[3]
> (2 (3 I E - Sqrt[3] E + 3 I E +
3/2 + I Sqrt[3]
> Sqrt[3] E )) /
I/2 Sqrt[3]
> ((-3 + I Sqrt[3]) (-3 I + Sqrt[3]) E ), 0},
12 E
> {0, 0, --------------------------------}}
(-3 I + Sqrt[3]) (3 I + Sqrt[3])
We anticipate that this error will be corrected for the next major release of Mathematica.
Additional Online Documentation:
Mathematica 3.0
http://documents.wolfram.com/v3/RefGuide/MatrixExp.html
Mathematica 4.0
http://documents.wolfram.com/v4/RefGuide/MatrixExp.html
Questions or comments? Send email to support@wolfram.com.
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