Monday, February 1, 2010

Matrix Exponential

We seek a solution of a homogeneous first order linear system of differential equations. For illustration purposes we consider the [Graphics:Images/MatrixExponentialMod_gr_1.gif] case:

[Graphics:Images/MatrixExponentialMod_gr_2.gif]
[Graphics:Images/MatrixExponentialMod_gr_3.gif]

First, write the system in vector and matrix form [Graphics:Images/MatrixExponentialMod_gr_4.gif]

[Graphics:Images/MatrixExponentialMod_gr_5.gif].

Then, find the eigenvalues and eigenvectors of the matrix [Graphics:Images/MatrixExponentialMod_gr_6.gif], denote the eigenpairs of A by

[Graphics:Images/MatrixExponentialMod_gr_7.gif] and [Graphics:Images/MatrixExponentialMod_gr_8.gif].

Assumption. Assume that there are two linearly independent eigenvectors [Graphics:Images/MatrixExponentialMod_gr_9.gif], which correspond to the eigenvalues [Graphics:Images/MatrixExponentialMod_gr_10.gif], respectively. Then two linearly independent solution to [Graphics:Images/MatrixExponentialMod_gr_11.gif] are

[Graphics:Images/MatrixExponentialMod_gr_12.gif], and
[Graphics:Images/MatrixExponentialMod_gr_13.gif].

Definition (Fundamental Matrix Solution) The fundamental matrix solution [Graphics:Images/MatrixExponentialMod_gr_14.gif], is formed by using the two column vectors [Graphics:Images/MatrixExponentialMod_gr_15.gif].

(1) [Graphics:Images/MatrixExponentialMod_gr_16.gif].

The general solution to [Graphics:Images/MatrixExponentialMod_gr_17.gif] is the linear combination

(2) [Graphics:Images/MatrixExponentialMod_gr_18.gif].

It can be written in matrix form using the fundamental matrix solution [Graphics:Images/MatrixExponentialMod_gr_19.gif] as follows

[Graphics:Images/MatrixExponentialMod_gr_20.gif].

Notation. When we introduce the notation

[Graphics:Images/MatrixExponentialMod_gr_21.gif],
and
[Graphics:Images/MatrixExponentialMod_gr_22.gif]

The fundamental matrix solution [Graphics:Images/MatrixExponentialMod_gr_23.gif] can be written as

(3) [Graphics:Images/MatrixExponentialMod_gr_24.gif].
or
(4) [Graphics:Images/MatrixExponentialMod_gr_25.gif].

The initial condition [Graphics:Images/MatrixExponentialMod_gr_26.gif]

If we desire to have the initial condition [Graphics:Images/MatrixExponentialMod_gr_27.gif], then this produces the equation

[Graphics:Images/MatrixExponentialMod_gr_28.gif].

The vector of constant [Graphics:Images/MatrixExponentialMod_gr_29.gif] can be solved as follows

[Graphics:Images/MatrixExponentialMod_gr_30.gif].

The solution with the prescribed initial conditions is

[Graphics:Images/MatrixExponentialMod_gr_31.gif].

Observe that [Graphics:Images/MatrixExponentialMod_gr_32.gif] where [Graphics:Images/MatrixExponentialMod_gr_33.gif] is the identity matrix. This leads us to make the following important definition

Definition (Matrix Exponential) If [Graphics:Images/MatrixExponentialMod_gr_34.gif] is a fundamental matrix solution to [Graphics:Images/MatrixExponentialMod_gr_35.gif], then the matrix exponential is defined to be

[Graphics:Images/MatrixExponentialMod_gr_36.gif].

Notation. This can be written as

(5) [Graphics:Images/MatrixExponentialMod_gr_37.gif],
or
(6) [Graphics:Images/MatrixExponentialMod_gr_38.gif].

Fact. For a [Graphics:Images/MatrixExponentialMod_gr_39.gif] system, the initial condition is

[Graphics:Images/MatrixExponentialMod_gr_40.gif],

and the solution with the initial condition [Graphics:Images/MatrixExponentialMod_gr_41.gif] is

[Graphics:Images/MatrixExponentialMod_gr_42.gif],
or
[Graphics:Images/MatrixExponentialMod_gr_43.gif].

Theorem (Matrix Diagonalization) The eigen decomposition of a [Graphics:Images/MatrixExponentialMod_gr_44.gif] square matrix A is

[Graphics:Images/MatrixExponentialMod_gr_45.gif],

which exists when A has a full set of eigenpairs [Graphics:Images/MatrixExponentialMod_gr_46.gif] for [Graphics:Images/MatrixExponentialMod_gr_47.gif], and d is the diagonal matrix

[Graphics:Images/MatrixExponentialMod_gr_48.gif]
and
[Graphics:Images/MatrixExponentialMod_gr_49.gif]

is the augmented matrix whose columns are the eigenvectors of A.

[Graphics:Images/MatrixExponentialMod_gr_50.gif].

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Introductory Methods of Numerical Analysis