Talk:Matrix exponential

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 $$e^{0M}=I$$. &mdash; 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. &mdash; 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)
 * I see this still hasn't been fixed. See  below.  — Arthur Rubin  (talk) 21:29, 23 October 2014 (UTC)

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)

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)

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)

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)

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)

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)

Commutativity
Does anybody know if the equation $$\exp(A+B)=\exp(A) \exp(B)$$ implies $$AB=BA$$? Franp9am (talk) 10:18, 20 June 2008 (UTC)


 * It does not. A counterexample (from Horn & Johnson) is given by
 * $$ A = \begin{bmatrix} 0 & 0 \\ 0 & 2\pi\textrm{i} \end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 0 & 1 \\ 0 & 2\pi\textrm{i} \end{bmatrix}. $$
 * $$A$$ and $$B$$ do not commute, but $$\exp(A+B) = \exp(A) \exp(B) = \text{the identity matrix}$$. I added something to the article. -- Jitse Niesen (talk) 10:59, 20 June 2008 (UTC)

- Note added; sorry if I'm not following conventions here, I haven't commented before. --Tom

It's also not even correct in general to say that $$\exp(A+B)=\exp(A)\exp(B)$$. For a simple counterexample, take nilpotent matrices $$A$$ and $$B$$ that sum to a nonsingular $$A+B$$, say

$$ A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix}. $$

The power series for $$\exp(A)=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$ and $$\exp(B)=\begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}$$ terminate after the second terms, but $$\exp(A+B)$$ never terminates at all: $$\exp(A+B)=\begin{bmatrix} \cos(1) & \sin(1) \\ -\sin(1) & \cos(1) \end{bmatrix}$$.

Horn and Johnson give a similar example in Topics in Matrix Analysis, p. 434-435. Automatic Tom (talk) 22:18, 9 August 2008 (UTC)


 * The above appears to be correct. Michael Hardy (talk) 21:46, 9 August 2008 (UTC)
 * Side-note: $$ A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix}$$ do not commute. Therefore they are not covered by the theorem. TN (talk) 11:55, 13 August 2020 (UTC)

The reference to the Horn and Johnson Topics in Matrix Analysis book quotes a theorem that says if $$ A $$ and $$ B $$ contain only algebraic entries then you can only say that $$ e^Ae^B = e^B\,e^A $$ if and only if $$ A $$ and $$ B $$ commute. This is different than saying $$ e^{A+B} = e^Ae^B $$ if and only if $$ A $$ and $$ B $$ commute. — Preceding unsigned comment added by Robleroble (talk • contribs) 08:23, 6 March 2013 (UTC)

Matrix Exponential via Laplace transform
I don't know how to properly code the math behind this, but the page is missing a section on solving for the matrix exponential via the Laplace transform.


 * $$ \mathcal{L}^{-1} \{ e^{At} \} = (sI-A)^{-1} $$

Substituting t=1 in the above equation will yield the matrix exponential of A.

99.236.42.178 (talk) 01:29, 3 November 2008 (UTC)


 * I am afraid this is correct but aggressively meaningless, as is section 2.5 which I propose to eliminate. That section is a circular restatement of true but jubilantly unusable facts. Unless you provided an explicit situation where a generalization of the Laplace transform for matrices provided explicit answers not available to the Taylor expansion of the exponential, or, in the truly astounding case, the Magnus expansion mentioned earlier, I don't see what the point of that section could possibly be.  Cuzkatzimhut (talk) 19:55, 1 November 2013 (UTC)


 * I'm not sure how this is "aggressively meaningless". If the point of section 2 is to give methods for the computation of the matrix exponential, then you can do it by the Laplace transform in just the way it was written. This isn't a generalization of the Laplace transform for matrices, merely the element-wise application of the inverse Laplace to $$(sI - A)^{-1}$$.  That being said, I don't know if this page should aim to provide every possible method for computing the matrix exponential. Zfeinst (talk) 01:54, 2 November 2013 (UTC)


 * OK, proceed to justify your statement by illustrating how the compact bottom line formula of the LT yields the above trivial formula of 2.4.1  for a general 2x2 matrix. It could make that section meaningful. As I indicated above, all this LT rigmarole is a circular way of rewriting the matrix exponential in terms of its power expansion, which is how it was defined in the first place, accepting application of the LT and its convergence peculiarities on the space of matrices. The connection to the differential equation is already detailed in section 1. You need a LT to handle an exponential? What would you believe the student of matrix exponentials would have lost, had she not read this subsection (2.5)? Cuzkatzimhut (talk) 11:55, 2 November 2013 (UTC)

I proceed to eliminate the LT tautology, and effectively assign its section number to the preceding subsection, 2.4.1, which evidently merits promotion to a self-standing section. I furthermore stuck the above s=1 formal identification in the Properties section further up in the article. Cuzkatzimhut (talk) 00:58, 5 November 2013 (UTC)

proof
It would be useful to sketch how some of the properties (eg. exp(M+N)=exp(M)exp(N) if ..) are proved. Cesiumfrog (talk) 07:59, 13 March 2010 (UTC)

Exponential of Sums Error & Lie Product Formula
I have a couple issues with this statement: "The converse is false: the equation eX + Y = eXeY does not necessarily imply that X and Y commute. However, the converse is true if X and Y contain only algebraic numbers and their size is at least 2×2 (Horn & Johnson 1991, pp. 435–437)"
 * "Horn & Johnson" should be omitted in favor of a link to the actual reference at the bottom. The reference is actually to "Topics in Matrix Analysis", whereas lots of people are used to thinking of "Matrix Analysis" when they see just "Horn & Johnson".
 * I read that section of Topics in Matrix Analysis this morning and what it actually says is that, if X and Y are NxN matrices, N >= 2, containing only algebraic numbers, then e^Xe^Y = e^Ye^X if and only if X and Y commute.  Nothing to do with e^{X+Y}.  It's also more of a side note in the H&J book; it's not proved or even stated as a theorem, lemma, or even a numbered equation.

On a related note, it may be helpful to some people (I know it would have been helpful to me!) to put a link somewhere here to the Lie product formula article, which says lim_{m\rightarrow\infty} (e^{A/m}e^{B/m})^m = e^{A+B} for ANY A and B.

I don't want to just make these changes because a) I don't really have time and b) I'm hoping someone who can take care to double check and integrate it with the rest of the article can do it.

--M0nstr42 (talk) 01:50, 30 September 2010 (UTC)

Exponential skew symmetric matrices
I added the section on skew symmetric matrices. My proofs are my own independent work - I have seen the results, however, given by a few different authors (only after I went through all the work did I bother to look anything up, derp). None of the sources prove my results. Does anyone know of anything more elegant than what I did? Lhibbeler (talk) 02:38, 4 November 2010 (UTC)

@ User:Arthur_Rubin, see Jean Gallier and Dianna Xu, "Computing Exponentials of Skew Symmetric Matrices And Logarithms of Orthogonal Matrices", International Journal of Robotics and Automation, 2000. My results are *not* original research. —Preceding unsigned comment added by Lhibbeler (talk • contribs) 04:19, 4 November 2010 (UTC)


 * Ahem. You said they were original research.  As for the polynomial equations for the exponentials of the skew-symmetric matrices, it's not that interesting a result.  It follows from some of the things which should be in [[matrix polynomial]; as a two-dimensional skew-symmetric matrix W has eigenvalues $$\pm i \alpha$$, and exp(W) has eigenvalues $$\exp(\pm i \alpha)$$.  Hence, if
 * $$P(x) = \cos \alpha + \frac {sin(\alpha)}{\alpha} x$$,
 * then P(W) = exp(W). — Arthur Rubin  (talk) 04:54, 4 November 2010 (UTC)


 * I said my proofs are original as far as I know- I have seen *the results* but not the derivations. I was trying to explain where the formulas come from rather than just throw up a formula. And who are you to say what is interesting? The results are interesting and important to mechanicians- using that formula is more than 10 times faster on my laptop than using Pade approximants. When you have a large number of skew matrices and you need to calculate their exponentials (rotation updates in crystal plasticity, for example), an order of magnitude decrease in computation time is, in fact, very interesting. It sounds like you have more beef with the matrix polynomial page than with my edit. If you still object to my revision, then get rid of the collapsible windows with the proofs. Besides, I have put two dents in the statement that opens the section "Finding reliable and accurate methods to compute the matrix exponential is difficult, and this is still a topic of considerable current research in mathematics and numerical analysis." Lhibbeler (talk) 05:19, 4 November 2010 (UTC)


 * You could list those formulas as examples in the "Alternative" section. Perhaps that section could be promoted, but, you really haven't provided anything else.  — Arthur Rubin  (talk) 05:49, 4 November 2010 (UTC)
 * Rechecking the 3D formula:
 * $$P(x) = 1 + \frac{\sin \alpha}{\alpha} x + \frac{1-\cos \alpha}{\alpha^2} x^2 $$
 * $$P(0) = 1 = \exp(0)$$
 * $$P(\pm i \alpha) = \cos \alpha \pm i \sin \alpha = \exp(\pm i \alpha)$$
 * Yep, it's the equation from the "Alternative" section. — Arthur Rubin  (talk) 05:55, 4 November 2010 (UTC)

What about this
For a 2x2 matrix A one can write the exponent as follows:


 * $$ e^A = \sqrt{e^{tr A}} (\frac{sinh(q)}{q}A + (cosh(q) - \frac{tr(A) sinh(q)}{2q}) I) $$

where
 * $$ q = \sqrt{-det(A-\frac{tr A}{2} I)} $$

and I is the 2x2 identity matrix, and sinh(0)/0 is set as 1.

This doesn't seem to be in the article and might be interesting for those looking for a closed expression. —Preceding unsigned comment added by 95.117.203.24 (talk) 12:15, 22 January 2011 (UTC)
 * Is that correct if A has its two eigenvalues equal, but is not a multiple of the identity? If it is, it seems notable, but is sufficiently complex as to require a source.  — Arthur Rubin  (talk) 15:07, 22 January 2011 (UTC)
 * Never mind, it's accurate. Still, a better expression might be:
 * $$ e^A = e^{\frac{\operatorname{tr} A}{2}} \left(\left(\cosh q\right) I +\frac{\sinh q }{q}\left(A-\frac{\operatorname{tr} A}{2}I\right) \right) $$
 * where
 * $$ q = \sqrt{-\det\left(A-\frac{\operatorname{tr} A}{2} I\right)} $$
 * Which can be seen by
 * $$e^\lambda \approx e^{\frac{\operatorname{tr} A}{2}}\left (\cosh q + \frac{\sinh q}{q} \left(\lambda - \frac{\operatorname{tr} A}{2}\right)\right) $$
 * at the eigenvalues &lambda; of A. — Arthur Rubin  (talk) 16:31, 22 January 2011 (UTC)


 * Hello. Yes, that case leads to q = 0.  I don't have a source for this equation but the proof is not hard:  one writes $$ A =  \begin{bmatrix}

a & b \\ c & d \end{bmatrix} = \begin{bmatrix} \frac{a+d}{2} & 0 \\ 0 & \frac{a+d}{2} \end{bmatrix} + \begin{bmatrix} \frac{a-d}{2} & b \\ c & \frac{d-a}{2} \end{bmatrix} = B + C $$. Since B and C commute we have e^A = (e^B)(e^C), and since B and C^2 are diagonal matrices, both e^B and e^C can be calculated directly.

Unfortunately this doesn't work so for 3x3 and bigger matrices. Maybe your reasoning will lead to a similar formula for bigger matrices, I'll have to think about it! —Preceding unsigned comment added by 95.117.203.24 (talk) 16:50, 22 January 2011 (UTC)


 * Included, now, as an alternative to the alternative section for 2&times;2 matrices. — Arthur Rubin  (talk) 17:22, 22 January 2011 (UTC)

Applications to quantum computing?
I didn't see anything in this article about its applications to quantum computing. In particular, with Hermitian matrices, Pauli operators, and rotations about the Bloch Sphere. There's a bunch about this in the book I'm reading: An Introduction to Quantum Computing (Phillip Kaye, Raymond Laflamme, and Michelle Mosca - 2007). Just thought I'd point this out in case anyone feels it's important to mention.

24.212.140.98 (talk) 21:55, 22 January 2012 (UTC)

A method I was taught, but cannot find external references
Hello, I was taught a method by a Hungarian mathematician and I'm not even sure what it is called. The problem is to find $$ e^{A t} $$. If you do not care about $$ t $$ then replace it with a 1 when done. I'm not sure if anyone else is aware of this method or if it is even useful. So I'm putting it here under the talk section of the article.

Anyways I will include an example and steps below. Given $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1/8 \\ 0 & 0 & 1/2 & 1/2 \end{pmatrix} $$

First, compute the eigenvalues. $$ \lambda_1 = 3/4 $$ and $$ \lambda_2 = 1 $$ each with an algebraic multiplicity of two.

Now, take the exponential of each eigenvalue multiplied by $$ t $$: $$ e^{\lambda_i t}$$. Multiply by an unknown matrix $$ B_i $$. If the eigenvalues have an algebraic multiplicity greater than 1, then repeat the process, but multiply by a factor of $$ t $$ for each repetition. If one eigenvalue had a multiplicity of three, then there would be the terms: $$ B_{i_1} e^{\lambda_i t}, B_{i_2} t e^{\lambda_i t}, B_{i_3} t^2 e^{\lambda_i t} $$. Sum all terms.

In our example we get:

$$ e^{A t} = B_{1_1} e^{\lambda_1 t} + B_{1_2} t e^{\lambda_1 t} + B_{2_1} e^{\lambda_2 t} + B_{2_2} t e^{\lambda_2 t} $$

$$ e^{A t} = B_{1_1} e^{3/4 t} + B_{1_2} t e^{3/4 t} + B_{2_1} e^{1 t} + B_{2_2} t e^{1 t} $$.

So how can we get enough equations to solve for all of the unknown matrices? Differentiate with respect to $$t$$.

$$ e^{A t} = B_{1_1} e^{3/4 t} + B_{1_2} t e^{3/4 t} + B_{2_1} e^{1 t} + B_{2_2} t e^{1 t} $$

$$ A e^{A t} = 3/4 B_{1_1} e^{3/4 t} + \left( 3/4 t + 1 \right) B_{1_2} e^{3/4 t} + 1 B_{2_1} e^{1 t} + \left(1 t + 1 \right) B_{2_2} e^{1 t} $$

$$ \begin{align} A^2 e^{A t} =& (3/4)^2 B_{1_1} e^{3/4 t} + \left( (3/4)^2 t + ( 3/4 + 1 \cdot 3/4) \right) B_{1_2} e^{3/4 t} + B_{2_1} e^{1 t} + \left(1^2 t + (1 + 1 \cdot 1 )\right) B_{2_2} e^{1 t} \\ =& (3/4)^2 B_{1_1} e^{3/4 t} + \left( (3/4)^2 t + 3/2 \right) B_{1_2} e^{3/4 t} + B_{2_1} e^{t} + \left(t + 2\right) B_{2_2} e^{t} \end{align} $$

$$ \begin{align} A^3 e^{A t} =& (3/4)^3 B_{1_1} e^{3/4 t} + \left( (3/4)^3 t + ( (3/4)^2 + (3/2) \cdot 3/4) ) \right) B_{1_2} e^{3/4 t} + B_{2_1} e^{1 t} + \left(1^3 t + (1 + 2) \cdot 1 \right) B_{2_2} e^{1 t} \\ =& (3/4)^3 B_{1_1} e^{3/4 t} + \left( (3/4)^3 t + 27/16 ) \right) B_{1_2} e^{3/4 t} + B_{2_1} e^{t} + \left(t + 3 \cdot 1 \right) B_{2_2} e^{t} \end{align} $$

Since the these equations must be true, regardless the value of $$t$$, we set $$t=0$$. Then we can solve for the unknown matrices.

$$ \begin{align} I =& B_{1_1} + B_{2_1} \\ A =& 3/4 B_{1_1} + B_{1_2} + B_{2_1} + B_{2_2} \\ A^2 =& (3/4)^2 B_{1_1} + (3/2) B_{1_2} + B_{2_1} + 2 B_{2_2} \\ A^3 =& (3/4)^3 B_{1_1} + (27/16) B_{1_2} + B_{2_1} + 3 B_{2_2} \end{align} $$

This can be solved using linear algebra (don't let the fact the variables are matrices confuse you). Once solved using linear algebra you have:

$$ \begin{align} B_{1_1} =& 128 A^3 - 366 A^2 + 288 A - 80 I \\ B_{1_2} =& 16 A^3 - 44 A^2 + 40 A - 12 I \\ B_{2_1} =&-128 A^3 + 366 A^2 - 288 A + 80 I\\ B_{2_2} =& 16 A^3 - 40 A^2 + 33 A - 9 I \end{align} $$

Plugging in the value for $$A$$ gives:

$$ \begin{align} B11 =& \begin{pmatrix}0 & 0 & 48 & -16\\ 0 & 0 & -8 & 2\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}\\ B12 =& \begin{pmatrix}0 & 0 & 4 & -2\\ 0 & 0 & -1 & \frac{1}{2}\\ 0 & 0 & \frac{1}{4} & -\frac{1}{8}\\ 0 & 0 & \frac{1}{2} & -\frac{1}{4}\end{pmatrix}\\ B21 =& \begin{pmatrix}1 & 0 & -48 & 16\\ 0 & 1 & 8 & -2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\\ B22 =& \begin{pmatrix}0 & 1 & 8 & -2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix} \end{align} $$

So the final answer would be:

$$ {e}^{t\,A}=\begin{pmatrix}{e}^{t} & t\,{e}^{t} & \left( 8\,t-48\right) \,{e}^{t}+\left( 4\,t+48\right) \,{e}^{\frac{3\,t}{4}} & \left( 16-2\,t\right) \,{e}^{t}+\left( -2\,t-16\right) \,{e}^{\frac{3\,t}{4}}\\ 0 & {e}^{t} & 8\,{e}^{t}+\left( -t-8\right) \,{e}^{\frac{3\,t}{4}} & -\frac{4\,{e}^{t}+\left( -t-4\right) \,{e}^{\frac{3\,t}{4}}}{2}\\ 0 & 0 & \frac{\left( t+4\right) \,{e}^{\frac{3\,t}{4}}}{4} & -\frac{t\,{e}^{\frac{3\,t}{4}}}{8}\\ 0 & 0 & \frac{t\,{e}^{\frac{3\,t}{4}}}{2} & -\frac{\left( t-4\right) \,{e}^{\frac{3\,t}{4}}}{4}\end{pmatrix} $$

This method was taught as it is nearly the same procedure to calculate $$A^t$$. Except that instead of taking the derivative to generate more equations, you can increment $$t$$ (which is useful for discrete difference equations). Also, replace using $$e^{(\lambda_i t)}$$ with $$\lambda_i^t$$. — Preceding unsigned comment added by 150.135.222.237 (talk) 18:57, 6 December 2013 (UTC)


 * I confess I don't know a name for it, and I have not stumbled on it in just this form in matrix analysis books. You might register to Wikipedia and clean it up and add it as a subsection near the end. It is quite tasteful in its immediacy, as it does not require diagonalizability, as manifest in your example of a neat defective matrix. The ansatz is a most general one for the generic polynomial dictated by the Cayley-Hamilton theorem, in your case a cubic; the "Wronskian" trick for multiple roots ensures linear independence, and is equivalent to the analysis of the Matrix_exponential subsection in the article. In fact, I wonder if you wished to adapt the example to a shorter section in the same notation, i.e. the ansatz exp(tA) = Bα exp (tα) + Bβ exp(tβ), yielding the same expression after solving this expression and its first derivative at t=0 for the Bs in terms A and I, arguably faster. It is a pretty method, and you'd be quite welcome to contribute it to the article (be bold, etc). Cuzkatzimhut (talk) 23:22, 15 December 2013 (UTC)
 * Actually, for distinct eigenvalues, it is just Sylvester's formula for the exponential, where the simplicity of the l.h.s. allows for ready indirect evaluation of the Bi s, the Frobenius covariants projecting onto the eigenspace corresponding to eigenvalues λi . For the general, non-diagonalizable case you are addressing, it is just about Buchheim's generalization. It' is well-worth covering in the article. Cuzkatzimhut (talk) 19:36, 16 December 2013 (UTC)


 * I am actually registered (I didn't sign in when I originally posted this), but following wikipedia's best practices, I cannot post "original research." Thus I cannot in good conscience post this as a subsection until someone can give this method a name or at least reference it elsewhere. Maybe this will help; this method is derived from the method of undetermined coefficients for linear differential equations. How to handle repeated roots (eigenvalues) follows from the same explanation here (https://www.khanacademy.org/math/differential-equations/second-order-differential-equations/complex-roots-characteristic-equation/v/repeated-roots-of-the-characteristic-equation). If you think this is enough support, then we (or I) can post it. Alternatively, if you can dig up the resources that discuss this being Bucheim's generalization of Sylvester's formula, then we can post it. .Mouse7mouse9 22:22, 29 December 2013 (UTC) — Preceding unsigned comment added by Mouse7mouse9 (talk • contribs)


 * I also wanted to mention there is a discrete variant (from solving discrete difference equations), allowing one to take symbolic matrix powers (raising a square matrix to an unknown scalar variable). I mention it here (https://en.wikipedia.org/wiki/Talk:Matrix_multiplication#Matrix_powers_and_discrete_difference_equations). Both the above method and the linked method are very similar. The symbolic matrix power is derived from solving discrete linear difference equations (again using the method of undetermined coefficients), rather than continuous differential equations, as in the case of the matrix exponential. — Preceding unsigned comment added by Mouse7mouse9 (talk • contribs) 23:26, 29 December 2013 (UTC)

I am sorry, I do not have a name for it, but, as I said, it is a less hidebound expression of 2.5. Different communities of users often employ different names. Please, do go ahead and post it--hopefully the names and the references will accrue later; but nobody would accuse you of posting original research, to the extent this is self-evident, a linear determination of the Frobenius covariants. So, for instance, you could call it subsection 2.6, something like: "A rapid implementation of Sylvester's formula", and work out first the trivial 2x2 example above, and then your neat defective matrix 4x4 example, in some semblance of uniform notation, as a segue to 2.5 (note its last case!). Ideally, somebody could provide a nice reference for it, but its logic is what is already covered in the exposition preceding it. I could help with the wp-linking after you were done. As an in itialreference, maybe this could do: Rinehart, R. F. (1955). "The equivalence of definitions of a matric function". The American Mathematical Monthly, 62 (6), 395-414. Cuzkatzimhut (talk) 00:03, 30 December 2013 (UTC)


 * OK, I put it in the page as stated. Lots of room for improvement, especially by the father thereof, User:Mouse7mouse9. A big thanks. Cuzkatzimhut (talk) 15:34, 30 December 2013 (UTC)

The new section: Rotation case
The new section Matrix exponential seems to be overly convoluted, and is unreferenced; I suspect that it is OR. It also semi-restricts itself to three dimensions, and uses perpendicular unit vectors unnecessarily. It also assumes Euclidean space without explicitly saying so – yet matrices and rotations apply in many contexts in which one does not (and in cases cannot) have an orthonormal basis. —Quondum 15:50, 25 September 2014 (UTC)
 * A simple rotation (which can be defined from a pair of vectors) in any number of dimensions should be expressible as a simple exponential $$R(\theta)=\exp(\theta G)$$ – no need for projections.
 * The "main article" Rodrigues' rotation formula is hardly appropriate. It is a way of handling rotation only in three dimensions involving quantities such as an axis of rotation that is not mentioned in this section, and is in a about sidestepping matrix exponentiation via vector algebra and trigonometry; thus hardly the subject of this article or this section.
 * The restriction to perpendicular unit vectors is unnecessary, though it does perhaps simplify normalization if one wishes to have only pure rotation (no dilations). A more general treatment (i.e. allowing any pair of vectors) should be hardly more complex.


 * I only cleaned it up and copyedited it, but it is anodyne and obvious... I have not seen the Weyl book adduced by the original author, but I trusted it is OK. In any case, the final link I adduced at the bottom of the section covers the same material, less abstractly. It cannot be OR, any more than the Pythagorean theorem can be!
 * The perpendicular vectors ensure the Projection operators work fine. You are right this is basically plane rotations embedded in an n-dim space, and all extra dimensions behave like 1 in the Rodrigues formula. Indeed, it deals with Euclidean space, and that fact should be thrown in, incidentally.
 * Indeed, any rotation is matrix exponential, as it is here, but the point is this exponential is explicitly calculated through the projection ops, which is why I suspect this subsection was put in this week. Perhaps the original author can argue for it. In any case calculation leads to the Rodrigues formula, and for the same reasons it has that form: the structure is evidently 3-dim.
 * I am befuddled by proposed avoidance of the Rodrigues formula. The answer is the Rodrigues formula, in its modern, matrix form, and it is, of course living effectively in 3d, as any direction perpendicular to the ab-plane is treated the same, and no rotations around different axes are composed. I think insistence on vector language for the Rodrigues is excessively antique: the Rodrigues formula is effectively the rotation matrix expressed as a quadratic polynomial in 3×3 spin matrices! The algebra is the same for n×n matrices, here, as asserted, and very different indeed from the generalization of the Rodrigues formula for higher dimensional representations of SO(3). I suspect any further discussion of this in the section cannot fail to confuse the reader, but if you had a pithy summary of the explanation, that would not be out of place.
 * I am not sure what you are proposing... This is not a deep geometrical point, it is the expansion of the exponential in powers of G with the obvious rearrangement of the self-evident structure. If the link of the last line is clearer, perhaps you might undertake to blend the two treatments. Cuzkatzimhut (talk) 17:01, 25 September 2014 (UTC)

PS on the clarification requested for the diagonalizable case: the exponential is but a sum of powers: a similarity transform of a power and of a sum of powers is but the same sum of the same powers of the similarity transform of the argument. Cuzkatzimhut (talk) 19:12, 25 September 2014 (UTC)

PSPS to ‎User:Jbergquist: it is best to leave questions here, rather than in comments in the text code. I had the same reaction myself, early on, yesterday, i.e. that maybe this belongs elsewhere, in one of the handful of articles on rotations. However, what is illustrated here is purely the technical evaluation of the matrix exponential, and it is perfectly position to segue from the previous, projector section, to the following one. This one illustrates the cyclic structure of G2=−I, i.e. that G behaves like i, superposed on the projector structure. It is a memorable derivation of the Rodrigues rotation formula, and illustrates its projective character. Cuzkatzimhut (talk) 20:44, 25 September 2014 (UTC)


 * It is probably best to keep the section as simple as possible. The definition of G comes from the derivation of R for a rotation of a vector in an arbitrary n-plane defined by a and b. Any two normal unit vectors in the plane will do. You get the same rotation matrix. A summary derivation of the exponential formula starting with the exponential expansion could be included. We could use a good reference for that. R(&theta;) = I cos(&theta;) + G sin(&theta;) only works for vectors in the plane so the projection operators are needed to split an arbitrary vector into two parts for the more general rotation matrix. The exponential expression in the definition of the rotation comes from a modification of plane rotation taking into account the Pab. My presentation here is somewhat backwards. The assumption is that the generator and rotation are derived elsewhere. The location indicated by the generator pipe would be a good place. I derived the results from scratch a short while ago and a simple method for finding a rotation matrix certainly deserves backup. I would be interested in hearing if you find this published elsewhere any time in the last 100 years or so. Sometimes mathematicians tend to be rather cryptic. --Jbergquist (talk) 22:14, 25 September 2014 (UTC)

I am in full agreement that this section should be kept breezy and short. It is self-explanatory to me, but then I am not the intended audience. I don't think more refs are needed, since, up to obvious changes of notation, it is a re-write of both WP links now provided at the end of the section. I think these links are enough. The subsection looks stable and informative.

To be more explicit as to why this is "essentially" SO(3): As it stands, of course it is an SO(2) rotation in n-dim space, i.e. generators of 2x2 blocks of a trivial nxn matrix. A further rotation, if in the same 2x2 block, would lead to trivial SO(2) composition. If in a completely disjoint 2x2 block, to an SO(2)xSO(2) structure. Only if it is in a 2x2 block overlapping this one, i.e. in a space involving, e.g. b and a different orthogonal c, but not a, do we have a nontrivial structure composing such rotations, to wit an SO(3), which makes the formula useful. (B.t.w., I would use the terminology "n-space" and "2-plane" embeddings in it, never "n-plane", but that's just personal taste.) You might take a look at Rotation_matrix to see what I mean. Cuzkatzimhut (talk) 22:59, 25 September 2014 (UTC) Cuzkatzimhut (talk) 22:59, 25 September 2014 (UTC)


 * Heh-heh. Do I see OR in progress?  Not that I'm against it, but don't ignore me when I point out missed opportunities for simplification. I'm liable to make certain statements that challenge some of the complexities and assumptions here.
 * Browsing the source (Weyl) did not show me a derivation as given using projections.
 * There is no "essentially SO(3)" about rotations in any number of dimensions, once the plane(s) of rotation are specified. Once a plane of rotation is fixed (only one plane in our example), it is "essentially SO(2)".
 * While projection is a neat way of deriving the result, it should produce a universal result in any number of dimensions: $R(θ) = exp(θG)$. This is easily seen: if the rotation is continuous, it can be subdivided into progressively smaller identical rotations that compose to the larger rotation.  Basically, given that the rotation matrix in any number of dimensions is $R(θ)$, we can find a matrix $G$ such that $R(θ) = lim_{n→∞}(I + θG/n)^{n} = exp(θG)$.
 * I'd actually be in favour of removing the projections in order to keep it short, simply stating the result, giving $G$ in terms of two vectors, though it would be just as easy to expand this to any number of mutually orthogonal vectors, paired to define planes, giving a general rotation in $n$ dimensions. —Quondum 00:16, 26 September 2014 (UTC)

I did not ignore you--- I did not imagine the author considered is as OR. Of course, as I keep indicating, it is not, as per the two WP pages linked at the end of the subsection. It is SO(3), though. You see the collapse to a trivial SO(2) in n=2, which does not tell anyone anything interesting, since it is the dumb rotation of a 2-vector not the Rodrigues formula. But the interesting part of the formula, as I indicate above, is precisely the SO(3), 3-plet representation (spin 1) part, hence the Rodrigues formula, precisely due to the projection. So this is the instructive part of the subsection. It is, of course, given in Axis–angle_representation but here it is produced compactly and directly. I did not appreciate your middle bullet: I cannot see how it has any bearing on the result, by which I mean an expansion of the exponential to be spanned by I, G and G2. This will not be the case for other reps of SO(3), such as spin 3/2, or generic SO(n) transformations, since higher powers of G would then survive in the expansion. My sense is that, as it stands, the subsection makes sense, and throwing the baby with the bathwater might exactly be not be salutary. Cuzkatzimhut (talk) 00:51, 26 September 2014 (UTC)
 * Sorry, I came across all wrong there. I did not intend to suggest that you did ignore me, only only that since I'm shooting from the hip, as it were, I may make several errors, but that there may be some thoughts worth sifting out of what I say. ;-)  —Quondum 04:37, 26 September 2014 (UTC)


 * On checking the expression for R(&theta;) I see that it is equal to eG&theta;. Thanks for pointing that out. I'll make the simplifications. --Jbergquist (talk) 01:54, 26 September 2014 (UTC)
 * There's a similar version of eG&theta; involving hyperbolic functions. The reference is Bjorken and Drell, Relativistic Quantum Mechanics, 1964, McGraw-Hill, p.22. --Jbergquist (talk) 02:25, 26 September 2014 (UTC)


 * I think a worked example may be needed. How about this?


 * $$\begin{align}

& a=\left( \begin{matrix}  1  \\   0  \\   0  \\ \end{matrix} \right)\quad b=\left( \begin{matrix}   0  \\   1  \\   0  \\ \end{matrix} \right) \\ & I=\left( \begin{matrix}  1 & 0 & 0  \\   0 & 1 & 0  \\   0 & 0 & 1  \\ \end{matrix} \right)\quad G=\left( \begin{matrix}   0 & -1 & 0  \\   1 & 0 & 0  \\   0 & 0 & 0  \\ \end{matrix} \right)\quad {{G}^{2}}=\left( \begin{matrix}   -1 & 0 & 0  \\   0 & -1 & 0  \\   0 & 0 & 0  \\ \end{matrix} \right) \\ & \\  & \theta =30^\circ \quad \Rightarrow \quad {{e}^{G\theta }}=\left( \begin{matrix}   0.866 & 0.5 & 0  \\   0.5 & 0.866 & 0  \\   0 & 0 & 1  \\ \end{matrix} \right) \\ \end{align}$$

--Jbergquist (talk) 03:40, 26 September 2014 (UTC)


 * Looking good. I still need to wrap my head around some of this. I've been transferring most of my thoughts from geometric algebra, and some of my inferences might not hold.  The translation from a rotor in GA to a rotation matrix is tricky. In particular, the equivalent of $G$ in GA, being $a ∧ b$, is generally invertible in the algebra in any dimension, unlike the matrix $G$. Something new for me. —Quondum 04:37, 26 September 2014 (UTC)
 * I was running checks on the formulas using Mathcad which doesn't check the mathematics behind the formulas. That's how the projection operators came to be involved and why the reduction to an exponential wasn't complete. I was bothered by that since the determinant |R(&theta;)| = 1 but I knew that the magnitude of the vector wasn't changed by the formulas posted. Mathtype is useful for doing examples but there's a sign missing above which I caught in time. --Jbergquist (talk) 04:51, 26 September 2014 (UTC)


 * I think I got off on the wrong foot by putting the cart before the horse which is why I decided to place the derivation of G in the generator article. I think initially we were working at cross purposes but I needed so show that the expression derived for R was related to an exponential function. That's why I chose to do it in this article and the projection operator P was needed for readers of the other article to connect R with eG&theta;. For n-dimensions matrix methods produce simpler results that tensors for rotations. Part of my motivation for doing all this stems from a lecture in the Mathematics department of a local university about 30 years ago in which it was mentioned that one of their students had claimed that you couldn't define an angle in four dimensions. (I collect Dover books too.) --Jbergquist (talk) 00:17, 28 September 2014 (UTC)
 * Do you have a particular interest in the claim? (It may be a confusion: assuming Euclidean space an angle between two vectors is defined in any number of dimensions, angles of rotation in 4 d are a well-defined composite of two independent angles in two fully orthogonal 2-d subspaces, and angles between planes are messy.) —Quondum 05:08, 28 September 2014 (UTC)

I tweaked the presentation to start from R=exp(θG) in the longer eqn, linking to the underlying point of the subsection, the exponential map involved in rotations, by wikilinking to the relevant section of rotation matrix; perhaps you'd choose to call that the "main article". And reorganized the expansion in powers of G to highlight the Rodrigues formula it is basically illustrating, without delving in a surfeit of notation.

My sense is that, here, no geometry or Lie theory is warranted---they are more than adequately covered in two handfuls (!) of relevant articles---but, instead, a compact illustration of evaluating simple matrix exponentials and the structure of the answers, and their relation to the evaluations of the more elaborate, non degenerate, real deal, matrix exponentials in the rest of the article. I sense that this has been achieved: Things appear stable now, but too much additional talk about rotations instead of just matrix exponentials might be out of place here. Cuzkatzimhut (talk) 10:37, 26 September 2014 (UTC)


 * I have no objection to your simplifications. We don't really need to talk about n-space at the beginning of the section. It's probably better left as a footnote at the end so we don't loose the majority of the readers. It's left out in the proof in the generator section. One should note that eG1&theta;1 and eG2&theta;2 don't commute so the order of their multiplication is significant when the two G's are not the same. This would be important for compound rotations where the matrix exponentials can't be added to produce a combined matrix exponential. --Jbergquist (talk) 21:06, 26 September 2014 (UTC)

I see I've unwittingly undone an intentional edit. Use of extra blank lines to partition text is a symptom of the need for more structure. In this case, the examples are pretty voluminous compared to the rest of the section. In this case, perhaps we need to place the examples in a collapsed expandable box? —Quondum 02:51, 27 September 2014 (UTC)

I've reduced the mention of higher dimensions to a footnote. The derivation of the expression for eG&theta; just depends on the antisymmetric properties of G which leads to the two dimensional properties of P and we can leave it at that. There's less clash present now. As is, article should be comprehensible to anyone with an elementary understanding of matrix algebra. --Jbergquist (talk) 20:43, 27 September 2014 (UTC)

Linear differential equation examples
We have a problem here. Neither of the recent versions satisfies $e^{0A} = I$. I don't have time to solve the equation at the moment. — Arthur Rubin (talk) 20:44, 23 October 2014 (UTC)
 * At least for the equation given right now (where the disputed tag is placed) it does satisfy $$e^{0A} = I$$. That doesn't necessarily mean this is the correct equation (I don't have the time to solve the equation at the moment either) but this quick complaint does not seem accurate to me. Zfeinst (talk) 21:38, 23 October 2014 (UTC)
 * I have now checked it, using the method of section 2.7: exp (tA) = B exp2t + Ct exp2t +D exp4t; it follows that B= A−A2/4,  D = 1−A +A2/4,  and C= −4 +3A −  A2/2. It all checks in the disputed formula, so I believe you should take down the unwarranted tag. Cuzkatzimhut (talk) 13:59, 24 October 2014 (UTC)

Matrix-matrix exponential
Why need the base be normal? As noted in matrix logarithm, any non-singular complex matrix has a logarithm.

— Although the paper states in this way, I actually agree with you that for the base matrix, just being non-singular suffice.Xuancong (talk) 03:17, 5 September 2015 (UTC)

Also, I would write $X^{Y}$ is similar to $^{Y}X$, rather than that they have the same eigenvalues. — Arthur Rubin (talk) 18:20, 4 September 2015 (UTC)

— Sure, you can change to 'similar' that if you wish.Xuancong (talk) 03:17, 5 September 2015 (UTC)

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Nomenclature
Please include explanation/definition/link for $$\oplus$$ and anything else not obvious to undergraduate mathematics students. —DIV (120.17.110.164 (talk) 12:48, 27 March 2018 (UTC))
 * Direct sum not obvious to math undergrads? Rather alarming. Cuzkatzimhut (talk) 21:08, 27 March 2018 (UTC)

Matrix-vector product
For iterative solvers (e.g. GMRES) one does not need to compute explicitly the matrix exponential $$e^M$$ but rather the action of the exponential on a vector: $$e^M \vec{b}$$. One can use Krylov subspace methods to approximate the action which is for large sparse systems more efficient.

Linking to existing packages such as http://www.maths.uq.edu.au/expokit/ would be helpful. HerrHartmuth (talk) 09:04, 6 December 2019 (UTC)