Matrix Exponential
- To: mathgroup at smc.vnet.net
- Subject: [mg34101] Matrix Exponential
- From: Kyriakos Chourdakis <k.chourdakis at qmul.ac.uk>
- Date: Sat, 4 May 2002 04:28:18 -0400 (EDT)
- Organization: Queen Mary, University of London
- Sender: owner-wri-mathgroup at wolfram.com
Dear all,
I am computing a few complex matrix exponentials for some Fourier
transforms, and I keep getting the same kind of result that I cannot
understand. As an example:
=============================
A) I can get the eigenvalues of the matrix.....
IN:
{{-1.850-1.993 \[ImaginaryI], -0.438-0.741
\[ImaginaryI]},{-0.00919+0.01696 \[ImaginaryI],-10.80+0.938
\[ImaginaryI]}} // Eigenvalues
OUT:
{-10.8017 + 0.937514 \[ImaginaryI], -1.84831 - 1.99251 \[ImaginaryI]}
=============================
B) I can also get the eigenvectors of the matrix......... and they are
apparently distinct.
IN:
{{-1.850-1.993 \[ImaginaryI], -0.438-0.741
\[ImaginaryI]},{-0.00919+0.01696 \[ImaginaryI],-10.80+0.938
\[ImaginaryI]}} // Eigenvectors
OUT:
{{0.0196355 + 0.0888622 \[ImaginaryI], 0.99585+ 0.\[ImaginaryI]},
{0.999998+0.\[ImaginaryI], -0.00148745 + 0.00140766 \[ImaginaryI]}}
=============================
C) But not the exponential, although they are related.
IN:
{{-1.850-1.993 \[ImaginaryI], -0.438-0.741
\[ImaginaryI]},{-0.00919+0.01696 \[ImaginaryI],-10.80+0.938
\[ImaginaryI]}} // MatrixExp
OUT:
Dot::"inf": "Input matrix contains an infinite entry."
Dot::"inf": "Input matrix contains an infinite entry."
\[Infinity]::"indet": "Indeterminate expression (0.+0.\[ImaginaryI])
ComplexInfinity encountered."
Dot::"mindet": "Input matrix contains an indeterminate entry."
Inverse::"inf": "Input matrix contains an infinite entry."
{{Indeterminate, Indeterminate},{0.00477109+
0.00196731\[ImaginaryI],0.000017144-0.0000126169\[ImaginaryI]}}.Inverse[{{ComplexInfinity,
0}, {0, 1}}]
=============================
I am not sure which one of the many dubious ways Mathematica uses for
its MatrixExp calculations.
Could someone enlighten me? I have observed that it happens as the
element differences become larger, although in the above example I would
not say they are that large. Is it that matrices need some rebalancing
before they are inserted in the MatrixExp[] function?
Best
Kyriakos.
_______________________________________
Kyriakos Chourdakis
Lecturer in Financial Economics
University of London
Queen Mary
London E1 4NS
URL: http://www.qmul.ac.uk/~te9001
Tel Wk: +44 207 7882 5086
Tel Mb: +44 793 140 1304
_______________________________________