Matrix - Jordan Normal/Canonical Form

**WWTL@WHL** · February 28th 2009, 06:19 PM

QUESTION:

Find $A^k$ when:

A = $\begin{bmatrix} -1 & -2 \\ 2 & 3 \end{bmatrix}$

__________________________________________________ ___

The eigenvalue (of multiplicity 2) is $x = 1$

I've tried putting A into Jordan-Normal/Canonical form, but I find that the matrix $(A- x I)^2 = 0$ , and I don't know what to do.

Please help.

**Soroban** · February 28th 2009, 07:03 PM

Hello, WWTL@WHL!

Find $A^k$ when: . $<br /> A \:= \: \begin{bmatrix} \text{-}1 & \text{-}2 \\ 2 & 3 \end{bmatrix}$

I see no choice but to crank out a few powers and seek a pattern . . .

. . $A \;=\;\begin{bmatrix}\text{-}1&\text{-}2\\2&3\end{bmatrix}$

. . $A^2 \;=\;\begin{bmatrix}\text{-}3 & \text{-}4 \\ 4 & 5 \end{bmatrix}$

. . $A^3 \;=\;\begin{bmatrix}\text{-}5&\text{-}6\\6&7\end{bmatrix}$

. . $A^4 \;=\;\begin{bmatrix}\text{-}7&\text{-}8\\8&9\end{bmatrix}$

. . $A^5 \;=\;\begin{bmatrix}\text{-}9&\text{-}10\\10&11\end{bmatrix}$

I think we've found the pattern . . .

We have: . $A^k \:=\:\begin{bmatrix}a&b\\c&d\end{bmatrix}$

. . where: $a$ is the negative of the $k^{th}$ odd number: . $-2k+1$

. . . . . . . $b$ is one less than $a\!:\;\;-2k$

. . . . . . . $c$ is the positive value of $b\!:\;\;2k$

. . . . . . . $d$ is one more than $c\!:\;\;2k+1$

Therefore: . $A^k \;=\;\begin{bmatrix}\text{-}2k+1 & \text{-}2k \\ 2k & 2k+1 \end{bmatrix}$

**Prometheus** · March 1st 2009, 10:18 PM

here is a more general way.

$det \begin{bmatrix} -1-x & \text{-}2 \\ 2 & 3-x \end{bmatrix}<br /> =(-1-x)(3-x)+4=x^2-2x+1=(x-1)^2$
so we need to find the canonical form. we can see that the matrix has only one eigenvector for 1, so the jordan form is
$J=P^{-1}AP=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$
to find P we can write P=[u,v] where u,v are vectors of size two and
$AP=PJ \rightarrow Av=v, Au=v+u$
so v is an eigenvector, for example (1,-1), and to find u solve the equation $(A-I)u=v$ .
now you have $A^k = (PJP^{-1})^k=PJ^k P^{-1}$
notice that this is a general way to find powers of matrices that have the jordan form. because this is used a lot, it is good to know what happens when you raise a jordan matrix to the power of k. to make it easier, write $J=dI+Z$ where the d is the eigenvector (in this case 1) and the Z matrix is a zero matrix with ones only above the main diagonal. in this case Z would just be $\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and then you have
$J^k = (dI+Z)^k = \sum \binom{k}{i} d^{k-i}Z^i$
and the thing about raising Z to the power of i is just just moving the "ones" i times to the top right corner, and here, because Z is only 2x2 matrix, moving the "ones" one time will take them out of the matrix, meaning $Z^2=0$ and because d=1 you get $J^k=\binom{k}{0}I+\binom{k}{1}Z=\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}$
this method works on the more general cases. for example, if you have more than one jordan block, then you can do this for every block separately.

**HallsofIvy** · March 2nd 2009, 06:33 AM

Originally Posted by WWTL@WHL

QUESTION:

Find $A^k$ when:

A = $\begin{bmatrix} -1 & -2 \\ 2 & 3 \end{bmatrix}$

__________________________________________________ ___

The eigenvalue (of multiplicity 2) is $x = 1$

I've tried putting A into Jordan-Normal/Canonical form, but I find that the matrix $(A- x I)^2 = 0$ , and I don't know what to do.

Please help.

Yes, of course $(A- I)^2= 0$ because every matrix satisfies its own characteristic equation.

First look for a eigenvectors for $\lambda= 1$ :
$\begin{bmatrix}-1 & -2 \\ 2 & 3\end{matrix}\begin{matrix} x \\ y\end{matrix}= \begin{matrix} x \\ y\end{bmatrix}$
which gives the equations -x- 2y= x and 2x+ 3y= y. Those both reduce to y= -x so <1, -1> is an eigenvector but we don't find a second. But because $(A- I)^2 v= 0$ for all vectors v and there are some such that (A- I)v is not 0, there must be v such that (A- I)((A-I)v)= 0 which means (A-I)v is an eigenvalue, a multiple of <1, -1>. That is, look for v such that $(A- I)v= \begin{bmatrix}-2 & -2 \\ 2 & 2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}1 \\ -1\end{bmatrix}$
That gives the equations -2x-2y= 1 and 2x+ 2y= -1 which are, again, equivalent. But taking x= 1 we get y= -3/2 so <1, -3/2> is a "generalized eigenvalue". Create the matrix $\begin{bmatrix}1 & 1 \\ -1 & -3/2\end{bmatrix}$ with those vectors as columns and find its inverse, $\begin{bmatrix}3 & 2 \\ -2 & -2\end{matrix}$

Now we have $\begin{bmatrix}3 & 2 \\ -2 & -2\end{bmatrix}\begin{bmatrix}-1 & -2 \\ 2 & 3\end{bmatrix}\begin{bmatrix}1 & 1 \\ -1 & -3/2\end{bmatrix}= \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}$ , the "Jordan Normal Form" for A.

That is, $P^{-1}AP= N$ so $A= PNP^{-1}$ and $A^n= PN^nP^{-1}$

And, of course, $N^n= \begin{bmatrix}1 & n \\ 0 1$ so to find $A^n$ , you just need to calculate $\begin{bmatrix}1 & 1 \\ -1 & -3/2\end{bmatrix}\begin{bmatrix}1 & n \\ 0 & 1\end{bmatrix}\begin{bmatrix}3 & 2 \\ -2 & -2\end{bmatrix}$ .

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