Talk:Matrix exponential

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[edit] Example (homogeneous)

Could someone check the result for the matrix exponential

e^{tM}=\begin{bmatrix} 2e^t - 2te^{2t} & -2te^{2t} & 0 \\ -2e^t + 2(t+1)e^{2t} & 2(t+1)e^{2t} & 0 \\ 2te^{2t} & 2te^{2t} & 2e^t\end{bmatrix}

The answer that Mathematica gives is quite different:

e^{tM}=\begin{bmatrix} \frac12 e^{2t}(1+e{2t}-2t) & -te^{2t} & e^{3t}\sinh t \\ -\frac12 e^{2t}(-1+e^{2t}-2t & (t+1)e^{2t} & -e^{3t}\sinh t \\ -\frac12 e^{2t}(-1+e^{2t}+2t & te^{2t} & e^{3t}\cosh t\end{bmatrix}

Obviously, this would give another solution to the system as well...

Well, the stated answer is wrong on its face, as e0M = I. — Arthur Rubin | (talk) 12:28, 22 August 2006 (UTC)
So, should we change it to the Mathematica-approved answer that I wrote a few lines above?
well, acutally, I get a different form:
e^(tM) = \begin{bmatrix} e^{3t} \cosh t - t e^{2t} & -t e^{2t} & e^{3t} \sinh t \\ -e^{3t} \sinh t + t e^{2t} & (t+1) e^{2t}& -e^{3t} \sinh t \\ e^{3t} \sinh t + t e^{2t} & t e^{2t} & e^{3t} \cosh t \end{bmatrix}
Same values, except for a couple of typos, but it looks simpler to me. — Arthur Rubin | (talk) 00:27, 31 August 2006 (UTC)
Perhaps it's better to use the same matrix that is mentioned further up in the article? -- Jitse Niesen (talk) 02:50, 31 August 2006 (UTC)

It looks like the two examples were copied from somewhere - it refers to examples "earlier in the article" where the exponential matrix is calculated, but this example doesn't exist. I can't tell if this is referring to a part of the wiki that was removed, or whether the example was just copied in it's entirety from another source. 130.215.95.157 (talk) 18:45, 5 March 2008 (UTC)

[edit] Column method

It seems to me that the column method is the same as the method based on the Jordan decomposition, but explained less clearly. Hence, I am proposing to remove that section. -- Jitse Niesen (talk) 20:11, 24 July 2005 (UTC)

I now removed the section. -- Jitse Niesen (talk) 11:21, 3 August 2005 (UTC)

I think there is indeed A Point to doing it like that, but I need to have a close look over it again as I'm a little unfamiliar on the material and need to get acquanted with it again, and I haven't had a chance to do this. On a cursory look the removal looks okay, though... Dysprosia 09:32, 4 August 2005 (UTC)

[edit] Continuity, etc.

For any two matrices X and Y, we have

\| e^{X+Y} - e^X \| \le \|X\| e^{\|X\|} e^{\|Y\|},

is clearly incorrect -- just look at the case X = 0. Perhaps the equation should be

\| e^{X+Y} - e^X \| \le e^{\|X\|} e^{\|Y\|},

(which I changed it to), but I'm not sure that's correct, either. Arthur Rubin | (talk) 21:12, 1 February 2006 (UTC)

Whoops, I'm pretty sure I put that in. According to H&J, Corollary 6.2.32, we have
\| e^{A+E} - e^A \| \le \|E\| e^{\|E\|} e^{\|A\|},
or, using X and Y,
\| e^{X+Y} - e^X \| \le \|Y\| e^{\|X\|} e^{\|Y\|}.
Apparently I made a mistake while renaming the variables. Thanks a lot! -- Jitse Niesen (talk) 21:36, 1 February 2006 (UTC)


[edit] Thank you

Hi, I just wanted to say thank you for writing this article so clearly. I needed to quickly look up how to do matrix exponentials again and I thought I'd try Wikipedia instead of MathWorld first this time. Nice simple explainations here, and written very clearly. This thanks also goes out to all deticated wikipedians who are updating the math pages.--Johnoreo 02:21, 9 February 2006 (UTC)

[edit] Arbitrary field?

The article mentions calculations over an arbitrary field. I think this should be changed since it gives the impression that there exists an exponential map over arbitrary fields.T3kcit (talk) 19:55, 16 December 2007 (UTC)

Agreed and changed. Solian en (talk) 14:37, 23 April 2008 (UTC)

[edit] Transition matrix

The transition matrix used here does not jive with what I understand is a matrix of eigenvectors. The rows need not sum to one. Is this a different matrix? If so, I haven't found an entry for the transition matrix used as a product with the diagonal matrix of eigenvalues and the inverse transition matrix to solve for the matrix exponential. The internal link does not clearly address this usage.John (talk) —Preceding comment was added at 02:31, 26 February 2008 (UTC)

The term transition matrix here means the matrix associated with the similarity transform that puts the matrix A into Jordan form; it has little to do with the meaning explained in transition matrix. I doubt this term is used very often, so I reformulated the text to avoid this term. Hope this helps. -- Jitse Niesen (talk) 14:52, 29 February 2008 (UTC)

[edit] Computing the matrix exponential (general case, arbitrary field)

There is no reference for the X = A + N decomposition over an arbitrary field. I couldn't find one in my textbooks (except for the Jordan decomposition in C). The French version points to the Dunford decomposition, which requires the matrix's minimal polynomial to have all its roots in the field. I believe that the decomposition does not hold in general. If no one objects, I will specify the conditions under which it exists. Solian en (talk) 15:04, 2 April 2008 (UTC)

Unfortunately, we've now obscured the fact that a complex matrix always has a unique A + N decomposition. I think we should drop consideration of arbitrary fields altogether until someone wants to handle them properly (probably in another section). At any rate it isn't at all clear when the article switches from the complex case to more general fields. -- Fropuff (talk) 16:30, 23 April 2008 (UTC)