matrix exponential

matrix exponential

The

exponential

of a real valued

square matrix

A

, denoted
by

e

A

, is defined as

e

A

=

∑

k

=

0

∞

1

k

!

⁢

A

k

=

I

+

A

+

1

2

⁢

A

2

+

⋯

Let us check that

e

A

is a real valued square matrix.
Suppose

M

is a

real number

such

|

A

i

⁢

j

|

<

M

for all
entries

A

i

⁢

j

of

A

.
Then

|

(

A

2

)

i

⁢

j

|

<

n

⁢

M

2

for all entries in

A

2

,
where

n

is the order of

A

. (Alternatively, one could argue using

matrix norms

: We have

||

e

A

||

≤

e

||

A

||

for the 2-norm, and hence the entries of

e

A

are

bounded

by

M

=

||

e

A

||

.) Thus,
in general, we have

|

(

A

k

)

i

,

j

|

<

n

k

⁢

M

k

+

1

. Since

∑

k

=

0

∞

n

k

k

!

⁢

M

k

+

1

converges

, we see that

e

A

converges to real valued

n

×

n

matrix.

Example 1.

Suppose

A

is

nilpotent

, i.e.,

A

r

=

0

for some

natural
number

r

. Then

e

A

=

I

+

A

+

1

2

!

⁢

A

2

+

⋯

+

1

(

r

-

1

)

!

⁢

A

r

-

1

.

Example 2.

If

A

is

diagonalizable

, i.e., of the form

A

=

L

⁢

D

⁢

L

-

1

, where

D

is a

diagonal matrix

, then

e

A

=

∑

k

=

0

∞

1

k

!

⁢

(

L

⁢

D

⁢

L

-

1

)

k

=

∑

k

=

0

∞

1

k

!

⁢

L

⁢

D

k

⁢

L

-

1

=

L

⁢

e

D

⁢

L

-

1

.

Further, if

D

=

diag

{

a

1

,

⋯

,

a

n

}

, then

D

k

=

diag

{

a

1

k

,

⋯

,

a

n

k

}

whence

e

A

=

L

⁢

diag

{

e

a

1

,

⋯

,

e

a

n

}

⁢

L

-

1

.

For

diagonalizable matrix

A

, it follows that

det

⁡

e

A

=

e

tr

A

.
However, this

formula

is, in fact, valid for all

A

.

Let

A

be a square

n

×

n

real valued matrix.
Then the

matrix exponential

satisfies

the following properties

1.

For the

n

×

n

zero matrix

O

,

e

O

=

I

, where

I

is the

n

×

n

identity matrix

.

2.

If

A

=

L

⁢

diag

{

a

1

,

⋯

,

a

n

}

⁢

L

-

1

for an

invertible

n

×

n

matrix

L

, then

e

A

=

L

⁢

diag

{

e

a

1

,

⋯

,

e

a

n

}

⁢

L

-

1

.

3.

If

A

and

B

commute,
then

e

A

+

B

=

e

A

⁢

e

B

.

4.

The trace of

A

and the

determinant

of

e

A

are related by the formula

det

⁡

e

A

=

e

tr

A

.

In effect,

e

A

is always invertible. The

inverse

is given by

(

e

A

)

-

1

=

e

-

A

.

5.

If

e

A

is a

rotational matrix

, then

tr

A

=

0

.

A relevant example on property 3.

We report an interesting example where the cited property is valid. In the field of

complex numbers

consider the

complex matrix

C

=

A

+

i

⁢

B

,

(1)

being

C

hermitian

, i.e.

C

⊺

=

C

¯

(here ”

⊺

” and overline ”

-

” stand for tranposition and conjugation, respectively) and

orthogonal

, i.e

C

-

1

=

C

⊺

. From (1),

C

⊺

=

A

⊺

+

i

⁢

B

⊺

.

Since

C

is orthogonal, from the

complex

equation

C

⁢

C

⊺

=

I

(

I

is the identity matrix), we have

C

⁢

C

⊺

=

(

A

+

i

⁢

B

)

⁢

(

A

⊺

+

i

⁢

B

⊺

)

=

(

A

⁢

A

⊺

-

B

⁢

B

⊺

)

+

i

⁢

(

B

⁢

A

⊺

+

A

⁢

B

⊺

)

=

I

,

whence the

imaginary part

leads to the equation

B

⁢

A

⊺

+

A

⁢

B

⊺

=

0

.

(2)

But

C

is also hermitian, so that

C

⊺

=

A

⊺

+

i

⁢

B

⊺

=

C

¯

=

A

-

i

⁢

B

,

therefore

A

⊺

=

A

is

symmetric

, and

B

⊺

=

-

B

is

skew-symmetric

. From these and (2),

B

⁢

A

=

A

⁢

B

, and this implies that

exp

⁡

(

A

)

⋅

exp

⁡

(

B

)

=

exp

⁡

(

A

+

B

)

. So that, the real and imaginary parts of an orthogonal and

hermitian matrix

verifies the property. Likewise, it is easy to show that if the complex matrix is symmetric and

unitary

, its real an

imaginary

components

also verify this property.

Title

matrix exponential

Canonical name

MatrixExponential

Date of creation

2013-03-22 13:33:27

Last modified on

2013-03-22 13:33:27

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

13

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15A15

Classification

msc 15-00

Related topic

ProofOfEquivalenceOfFormulasForExp

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