$\displaystyle e^A=Xe^DX^{-1}$

$\displaystyle A=\begin{bmatrix}

1 & 1\\

-1 & -1

\end{bmatrix}$

$\displaystyle \lambda_1=\lambda_2=0$

This will yield only 1 eigenvector so this matrix can't be diagonlaized.

How can I do this problem?

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- Apr 25th 2010, 01:20 PMdwsmithmatrix exponential
$\displaystyle e^A=Xe^DX^{-1}$

$\displaystyle A=\begin{bmatrix}

1 & 1\\

-1 & -1

\end{bmatrix}$

$\displaystyle \lambda_1=\lambda_2=0$

This will yield only 1 eigenvector so this matrix can't be diagonlaized.

How can I do this problem? - Apr 25th 2010, 01:54 PMMoo
Hello,

Actually, $\displaystyle e^A=\sum_{k=0}^\infty \frac{1}{k!} \cdot A^k=I+A+\sum_{k=2}^\infty \frac{1}{k!} \cdot A^k$

but if you look at A, you'll see that $\displaystyle A^2=\begin{pmatrix}0&0\\0&0\end{pmatrix}$

so $\displaystyle A^k=\begin{pmatrix}0&0\\0&0\end{pmatrix} ~,~ \forall k\geq 2$

finally, $\displaystyle e^A=I+A$ :) - Apr 25th 2010, 02:00 PMdwsmith
Is there a way to do this with diagonal matrices like first wrote?

- Apr 25th 2010, 02:10 PMMoo
No.

And I showed you this because the method of diagonalizing comes from this formula.

If one can diagonalize the matrix A, then we have $\displaystyle A=XDX^{-1}$, where D is a diagonal matrix.

So $\displaystyle A^k=XD^kX^{-1}$, and $\displaystyle D^k$ is always easy to compute. That's why it is set this way almost every time you deal with the exponential of a matrix.

If you want to see how it can be used with a diagonalizable matrix, tell me, I'll find a previous thread in this forum for ya - Apr 25th 2010, 02:13 PMdwsmith
Then neither of these matrices can be done via diagonalization:

$\displaystyle \begin{bmatrix}

1 & 1\\

0 & 1

\end{bmatrix}$ and $\displaystyle \begin{bmatrix}

1 & 0 & -1\\

0 & 1 & 0\\

0 & 0 & 1

\end{bmatrix}$ then either, correct? - Apr 25th 2010, 02:16 PMMoo
If I'm not mistaking, they're both diagonalizable, so it can be done with the method you want so much to use :D

(for both of them, the corresponding diagonal matrix should be the identity matrix) - Apr 25th 2010, 02:18 PMdwsmith
- Apr 25th 2010, 02:24 PMMoo
For the first one, 1 is an eigenvalue with multiplicity 2, and for the second one, 1 is an eigenvalue with multiplicity 3.

The eigenspace related to 1 is respectively of dimension 2 and 3.

There's no doubt about it, use characteristic polynomials :D - Apr 25th 2010, 02:29 PMdwsmith
$\displaystyle \begin{bmatrix}

1-\lambda & 1\\

0 & 1-\lambda

\end{bmatrix}\Rightarrow\begin{bmatrix}

0 & 1\\

0 & 0

\end{bmatrix}\Rightarrow x_1\begin{bmatrix}

1 \\

0

\end{bmatrix}$

I only have one egienvector. What went wrong? - Apr 26th 2010, 03:35 AMHallsofIvy
No, neither is diagonalizable. The first is already in "Jordan Normal Form" and the other can be put in that form.

As you say, "1" is the only eigenvalue and <1, 0, 0> and <0, 1, 0> span the two-dimensional eigenspace.

Ir can be put in Jordan Normal Form $\displaystyle N= \begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$ which would make it slightly easier to find powers or the exponential of the matrix (though not as easy as if it were diagonal). For example,

$\displaystyle N^2= \begin{bmatrix}1 & 2 & 0 \\ 0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$

$\displaystyle N^3= \begin{bmatrix}1 & 3 & 0 \\ 0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$

$\displaystyle N^4= \begin{bmatrix}1 & 4 & 0 \\ 0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$

and, in general,

$\displaystyle N^k= \begin{bmatrix}1 & k & 0 \\ 0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$

$\displaystyle e^N= $$\displaystyle \begin{bmatrix}1+ 1+ 1/2+ 1/6+...+ 1/k!+ ...& 1+ 1+ 2/2+ 3/6+ ...k/k!+ ... & 0\end{bmatrix}$$\displaystyle \begin{bmatrix} 0 & 1+ 1+ 1/2+ 1/6+ ...+ 1/k!+... & 0 \\ 0 & 0 & 1+ 1+ 1/2+ 1/6+ ...+ 1/k!+ ...\end{bmatrix}$

(Sorry about the break in the matrix, but the original was too large to fit the LaTex.)

Note that 1+ 1+ 2/2+ 3/6+ ...+ k/k!+ ...= 1+ (1+ 1+ 1/2+ ...+ 1/(k-1)!+ ... = 1+ e.

That is, $\displaystyle e^N= \begin{bmatrix}e & 1+ e & 0 \\ 0 & e & 0\\ 0 & 0 & e\end{bmatrix}$. - Apr 27th 2010, 11:44 AMMoo
Owww... Sorry :( I promise I won't reply to any linear/abstract algebra for a long time ! (not a big loss :D)