Inside Collection (Course): Matrix Analysis

Summary: This module introduces how to compute the matrix exponential using eigenvalues and eigenvectors.

In this module we exploit the fact that the matrix exponential
of a diagonal matrix is the diagonal matrix of element
exponentials. In order to exploit it we need to recall that all
matrices are almost diagonalizable. Let us begin with the clean
case: if

If

As a second example let us suppose

```
>> [S, Lam] = eig(A)
S = 0.7071 0.7071
0 + 0.7071i 0 - 0.7071i
Lam = 0 + 1.0000i 0
0 0 - 1.0000i
>> Si = inv(S)
Si = 0.7071 0 - 0.7071i
0.7071 0 + 0.7071i
>> simple(S*diag(exp(diag(Lam)*t))*Si)
ans = [ cos(t), sin(t)]
[-sin(t), cos(t)]
```

If

```
>> [S, Lam] = eig(A)
S = 1.0000 -1.0000
0 0.0000
Lam = 0 0
0 0
```

So zero is a double eigenvalue with but one eigenvector. Hence

Conversely we express the resolvent
*ALL* of the eigenvalues
of

```
>> R = inv(s*eye(3)-A)
R = [ 1/(s-1), 1/(s-1)^2, 0]
[ 0, 1/(s-1), 0]
[ 0, 0, 1/(s-2)]
```

and so

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