Question

Having trouble with Example 4 from page 320. I'll post the full example and my specific question below.

Answer 1

I understand everything up until the part I highlighted. Specifically I do not understand how \[(A-I)^{2}\] gives the zero matrix.

Also, I thought the definition of \[e^{At}\] as a series would look something like
\[I + At + \frac{1}{2}(At)^{2} + \frac{1}{6}(At)^{3} + ... \]
As shown in equation 14 on page 319.

I don't understand how factoring \[e^{At}\] into \[e^{It}e^{(A-I)t}\] changes the computation of its definition as a series.

This is the first chapter where I feel utterly lost in some of the examples. :(

Answer 2

In response to your first question, the example defines A as the coefficient matrix of y' and -y+2y', so that
\[A=\left[\begin{smallmatrix}0&1\\-1&2\end{smallmatrix}\right]\\
A-I=\left[\begin{smallmatrix}-1&1\\-1&1\end{smallmatrix}\right]\]
Which if you square is the zero matrix.
In response to your second question, I don't have your book in front of me (I'm not taking this course), but I think what they're trying to say is the following, and remember that (A-I)^2 = 0:
\[\begin{align}
e^{At} &= e^{It-It+At}\\
&=e^{It}e^{At-It}\\
&=e^{It}e^{(A-I)t}\\
&=e^t\left(I+(A-I)t + \frac{((A-I)t)^2}{2} + \frac{((A-I)t)^3}{6} + \frac{((A-I)t)^4}{24}+\cdots\right)\\
&=e^t\left(I+(A-I)t + \frac{(A-I)^2t^2}{2} + \frac{(A-I)^3t^3}{6} + \frac{(A-I)^4t^4}{24}+\cdots\right)\\
&=e^t\left(I+(A-I)t + \frac{(A-I)^2t^2}{2} + \frac{(A-I)^3t^3}{6} + \frac{(A-I)^4t^4}{24}+\cdots\right)\\
&=e^t\left(I+(A-I)t + \frac{0\cdot t^2}{2} + \frac{0\cdot(A-I)\cdot t^3}{6} + \frac{0\cdot (A-I)^2\cdot t^4}{24}+\cdots\right)\\
&=e^t\left(I+(A-I)t + 0 + 0 + 0 + \cdots\right)\\
&=e^t\left(I + (A-I)t\right)
\end{align}\]
Which is then easy enough to solve.

Answer 3

Top notch. That makes it a lot more clear! That was an excellent explanation! I'm still wrestling with what exactly is a matrix exponent, but I'm going to push ahead and works some problems and then maybe it will seem more clear.

Thank you!

Answer 4

My pleasure. If you need any more help with linear algebra (or abstract algebra in general) just give a yell.