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##### Lenses

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• Lens for Engineering

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By: Sidney Burrus

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Inside Collection (Course):

Course by: Steven J. Cox. E-mail the author

# Overview

Module by: Steven J. Cox. E-mail the author

Summary: This module describes the general idea of matrix exponential.

The matrix exponential is a powerful means for representing the solution to nn linear, constant coefficient, differential equations. The initial value problem for such a system may be written xt=Axt x t A x t x0= x0 x 0 x0 where AA is the n-by-n matrix of coefficients. By analogy to the 1-by-1 case we might expect

xt=eAtu x t A t u
(1)
to hold. Our expectations are granted if we properly define eAt A t . Do you see why simply exponentiating each element of At will no suffice?

There are at least 4 distinct (but of course equivalent) approaches to properly defining eAt A t . The first two are natural analogs of the single variable case while the latter two make use of heavier matrix algebra machinery.

Please visit each of these modules to see the definition and a number of examples.

For a concrete application of these methods to a real dynamical system, please visit the Mass-Spring-Damper module.

Regardless of the approach, the matrix exponential may be shown to obey the 3 lovely properties

1. ddteAt=AeAt=eAtA t A t A A t A t A
2. eA( t1 + t2 )=eA t1 eA t2 A t1 t2 A t1 A t2
3. eAt A t is nonsingular and eAt-1=e(At) A t A t

Let us confirm each of these on the suite of examples used in the submodules.

## Example 1

If A=( 10 02 ) A 1 0 0 2 then eAt=( et0 0e2t ) A t t 0 0 2 t

1. ddteAt=( et0 02e2t )=( 10 02 )( et0 0e2t ) t A t t 0 0 2 2 t 1 0 0 2 t 0 0 2 t
2. ( e t1 + t2 0 0e2 t1 +2 t2 )=( e t1 e t2 0 0e2 t1 e2 t2 )=( e t1 0 0e2 t1 )( e t2 0 0e2 t2 ) t1 t2 0 0 2 t1 2 t2 t1 t2 0 0 2 t1 2 t2 t1 0 0 2 t1 t2 0 0 2 t2
3. eAt-1=( et0 0e(2t) )=e(At) A t t 0 0 2 t A t

## Example 2

If A=( 01 -10 ) A 0 1 -1 0 then eAt=( costsint sintcost ) A t t t t t

1. ddteAt=( sintcost costsint ) t A t t t t t and AeAt=( sintcost costsint ) A A t t t t t
2. You will recognize this statement as a basic trig identity ( cos t1 + t2 sin t1 + t2 sin t1 + t2 cos t1 + t2 )=( cos t1 sin t1 sin t1 cos t1 )( cos t2 sin t2 sin t2 cos t2 ) t1 t2 t1 t2 t1 t2 t1 t2 t1 t1 t1 t1 t2 t2 t2 t2
3. Using the cofactor expansion we find eAt-1=( costsint sintcost )=( costsint sintcost )=e(At) A t t t t t t t t t A t

## Example 3

If A=( 01 00 ) A 0 1 0 0 then eAt=( 1t 01 ) A t 1 t 0 1

1. ddteAt=( 01 00 )=AeAt t A t 0 1 0 0 A A t
2. ( 1 t1 + t2 01 )=( 1 t1 01 )( 1 t2 01 ) 1 t1 t2 0 1 1 t1 0 1 1 t2 0 1
3. ( 1t 01 )-1=( 1t 01 )=e(At) 1 t 0 1 1 t 0 1 A t

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#### Collection to:

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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks