# 0.13 Matrix exponential

 Page 1 / 1
A formal description of the matrix exponential. The definition is given as well as examples of calculating it. The matrix exponential in thefrequency domain is given as well through treatment by the Laplace transform.

Since systems are often represented in terms of matrices and solutions of system equations often make use of the exponential, it makes senseto try and understand how these two concepts can be combined. In many previous applications, we've seen terms like $e^{at}$ come in handy for talking about system behavior. Here, awas always a scalar quantity. However, what would happen if the scalar $a$ was replaced by a matrix $A$ ? The result would be what is known as the matrix exponential .

## Definition

Recall the definition of the scalar exponential:

$e^{at}=1+a\frac{t}{1!}+a^{2}\frac{t^{2}}{2!}+a^{3}\frac{t^{3}}{3!}+$

The definition of the matrix exponential is almost identical:

$e^{At}={I}_{n}+A\frac{t}{1!}+A^{2}\frac{t^{2}}{2!}+A^{3}\frac{t^{3}}{3!}+$

Where $A$ is $n$ x $n$ and ${I}_{n}$ is the $n$ x $n$ identity matrix. While it is nice to see the resemblance between these two definitions, applying this infinite series does not turnout to be very efficient in practice. However, it can be useful in certain special cases.

Compute $e^{At}$ where $A=\begin{pmatrix}0 & 1\\ -1 & 0\\ \end{pmatrix}$ . We can start by taking powers of $A$ so that we can use the formal definition.

$A=\begin{pmatrix}0 & 1\\ -1 & 0\\ \end{pmatrix}$
$A^{2}=\begin{pmatrix}0 & 1\\ -1 & 0\\ \end{pmatrix}\begin{pmatrix}0 & 1\\ -1 & 0\\ \end{pmatrix}=\begin{pmatrix}-1 & 0\\ 0 & -1\\ \end{pmatrix}=-I$
$A^{3}=A^{2}A=-A$
$A^{4}=A^{2}A^{2}=I$
$A^{5}=AA^{2}=A$
$A^{6}=A^{2}A^{4}=-I$

And so the pattern goes, giving:

$A^{(4(n-1)+1)}=A$
$A^{(4(n-1)+2)}=-I$
$A^{(4(n-1)+3)}=-A$
$A^{(4(n-1)+4)}=I$

If we fill in the terms in the definition of $e^{at}$ , we'll get the following matrix:

$e^{At}=\begin{pmatrix}1-\frac{t^{2}}{2!}+\frac{t^{4}}{4!}- & t-\frac{t^{3}}{3!}+\frac{t^{5}}{5!}-\\ -t+\frac{t^{3}}{3!}-\frac{t^{5}}{5!}+ & 1-\frac{t^{2}}{2!}+\frac{t^{4}}{4!}-\\ \end{pmatrix}$

We notice that the sums in this matrix look familiar-in fact, they are the Taylor Series expansions of the sinusoids.Therefore, the solution further reduces to:

$e^{At}=\begin{pmatrix}\cos t & \sin t\\ -\sin t & \cos t\\ \end{pmatrix}$

## General method

The example above illustrates how the use of the true definition to simplify matrix exponentials might only be easily applied in caseswith inherent repetition. There is a more general method involving the Laplace Transform. In particular,

$(e^{At})=sI-A^{(-1)}$

We can verify that this is true by inserting the formal definition of the matrix exponential:

$(e^{At})=(I+A\frac{t}{1!}+A^{2}\frac{t^{2}}{2!}+)=\frac{1}{s}I+\frac{1}{s^{2}}A+\frac{1}{s^{3}}A^{2}+=sI-A^{(-1)}$

The jump between the third and fourth equations here may be a bit hard to believe, but this equality reduces to $I=I$ when both sides are multiplied by $sI-A$ . Taking an inverse Laplace of each side of Laplace Transform of the equation we find an expression for the matrix exponential:

$e^{At}=^{(-1)}(sI-A^{(-1)})$

We can do the same example as before, this time using the Laplace-based method.

$A=\begin{pmatrix}0 & 1\\ -1 & 0\\ \end{pmatrix}$
$sI-A^{(-1)}=\begin{pmatrix}s & -1\\ 1 & s\\ \end{pmatrix}^{(-1)}=\frac{1}{s^{2}+1}\begin{pmatrix}s & 1\\ -11 & s\\ \end{pmatrix}=\begin{pmatrix}\frac{s}{s^{2}+1} & \frac{1}{s^{2}+1}\\ \frac{-1}{s^{2}+1} & \frac{s}{s^{2}+1}\\ \end{pmatrix}$

Taking the inverse laplace of this gives us

$e^{At}=\begin{pmatrix}\cos t & \sin t\\ -\sin t & \cos t\\ \end{pmatrix}$

## Properties of the matrix exponential

In the scalar case, a product of exponentials $e^{a}e^{b}$ reduces to a single exponential whose power is the sum of the individual exponents' powers, $e^{a+b}$ . However, in the case of the matrix exponential, this is nottrue. If $A$ and $B$ are matrices,

$e^{A}e^{B}\neq e^{A+B}$

unless $A$ and $B$ are commutative (i.e. $AB=BA$ )

The derivative operates on the matrix exponential the same as it does on the scalar exponential.

$\frac{d e^{At}}{d t}}=0+A+A^{2}\frac{t}{1!}+A^{3}\frac{t^{2}}{2!}+=A(I+A\frac{t}{1!}+A^{2}\frac{t^{2}}{2!}+)=Ae^{At}$

#### Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, State space systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10143/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'State space systems' conversation and receive update notifications?

 By Brooke Delaney By OpenStax By By Richley Crapo By Anonymous User By OpenStax By OpenStax By Savannah Parrish By OpenStax By Abby Sharp