Matrix exponentials

Defining the exponential of a matrix by its power series, using it to write the solution and a fundamental matrix at once, and computing it by diagonalization.

Notes on Diffy Qs: Differential Equations for Engineers, Matrix exponentials

Status

FlagsLoading your progress…

5 exercises

Exercise 101

Compute $e^{tA}$ where $A=\left[ \begin{smallmatrix} 1 & -2 \\ -2 & 1 \end{smallmatrix}\right]$ .

2.5

Exercise 102

Compute $e^{tA}$ where $A=\left[ \begin{smallmatrix} 1 & -3 & 2 \\ -2 & 1 & 2 \\ -1 & -3 & 4 \end{smallmatrix}\right]$ .

2.5

Exercise 103

Compute $e^{tA}$ where $A=\left[ \begin{smallmatrix} 3 & -1 \\ 1 & 1 \end{smallmatrix}\right]$ .
Solve $\vec{x}\,' = A \vec{x}$ for $\vec{x}(0) = \left[ \begin{smallmatrix} 1 \\ 2 \end{smallmatrix}\right]$ .

2.5

Exercise 104

Compute the first 3 terms (up to the second degree) of the Taylor expansion of $e^{tA}$ where $A=\left[ \begin{smallmatrix} 2 & 3 \\ 2 & 2 \end{smallmatrix}\right]$ . Write it as a single matrix. Then use it to approximate $e^{0.1A}$ .

2.5

Exercise 105

For any positive integer $n$ , find a formula (or a recipe) for $A^n$ for the following matrices:

$\begin{bmatrix} 7 & 4 \\ -5 & -2 \end{bmatrix}$
$\begin{bmatrix} -3 & 4 \\ -6 & 7 \end{bmatrix}$
$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

2.5