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Let $$A = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right )$$

What can I say about the matrix $A^k$ for arbitrary natural $k$. Is there some way to express $A^k$ in terms of $A$. I have observed that $A^4 = I$.

EDIT: I want a better form of $A^k$ in order to find a closed form expression in terms of elementary functions for $\exp(A)$

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I believe that it is difficult to get one single expression without invoking something like either $\sin\left(\frac{k\cdot\pi}{2}\right)$ or $\mathop{Re}(i^k)$ –  Arthur Feb 5 '13 at 15:02
    
Write down the powers of $A$. What do you notice? –  Ben Millwood Feb 5 '13 at 15:06
    
calculate eigen value and vector value then write diagonal form of A simply calculate $A^k$.... –  Maisam Hedyelloo Feb 5 '13 at 15:08
    
Unfortunately I don't notice something other than what I have already stated. –  user44069 Feb 5 '13 at 15:09
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3 Answers

up vote 6 down vote accepted

More generally, if $$B(\theta) = \left( \begin{array}{cc} \cos \theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array} \right )$$

Then $$B(\theta)^n=\left( \begin{array}{cc} \cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \\ \end{array} \right ) = B(n\theta)$$

Your case is $A=B(\frac\pi 2)$.

In that particular case:

$$A^n = \begin{cases}I & \text{if }n\equiv 0 \pmod 4\\A&\text{if }n\equiv 1\pmod 4\\-I &\text{if }n\equiv 2\pmod 4\\-A&\text{if }n\equiv 3\pmod 4\end{cases}$$

Writing out the terms for $\exp(A)$, we get:

$$\exp(A) = (I-I^2/2!+I^4/4!-...) + A(I/1!-I/3!+I/5!...) = \\ \left( \begin{array}{cc} \cos 1 & 0 \\ 0 & \cos 1 \\ \end{array} \right )+A\left( \begin{array}{cc} \sin 1 & 0 \\ 0 & \sin 1 \\ \end{array} \right )=\\\left( \begin{array}{cc} \cos 1 & -\sin 1 \\ \sin 1 & \cos 1 \\ \end{array} \right )$$

More generally, $$\exp(B(\theta)) =\left( \begin{array}{cc} e^{\cos\theta}\cos(\sin\theta) & -e^{\cos\theta}\sin(\sin\theta) \\ e^{\cos\theta}\sin(\sin\theta) & e^{\cos\theta}\cos(\sin\theta) \\ \end{array} \right )$$

Fundamentally, what is going on here is that the matrices of the form $$\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right )=aI + bA, a,b\in\mathbb R$$ form a sub-algebra of the algebra of matrices. This sub-algebra is completely isomorphic[*] to the complex numbers, with $A$ corresponding to $i$. In particular, then $\exp(A)$ corresponds to $\exp(i)=cos 1 + i\sin 1$ which corresponds to $cos 1 + (\sin 1) A$

[*] I am using this term, "completely isomorphic," loosely, but basically, I mean that not only are the algebras the same, but also they are topologically equivalent - convergence of series in one algebra implies convergence of the corresponding series in the other algebra, for example.

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I should have stated from the beginning that I want a better form of $A^k$ in order to find a closed form expression in terms of elementary functions for $\exp(A)$ –  user44069 Feb 5 '13 at 15:16
    
I think for exponentiation, it would be easier to diagonalize $A=B\Lambda B^{-1}$ for some $B$ and $\Lambda$ a diagonal matrix with the eigenvalues $i$ and $-i$. Thn $exp(A)=B\exp(\Lambda)B^{-1}$ –  Thomas Andrews Feb 5 '13 at 15:21
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We can diagonalize $A$ by calculating the Jordan Normal Form, using it's eigenvalues, $\lambda_{1, 2} = \pm i$, where $i$ is the imaginary unit for this matrix as:

$$A = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right ) = S \cdot J \cdot S^{-1} = \left( \begin{array}{cc} -i & \ i \\ 1 & 1 \\ \end{array} \right ) \left( \begin{array}{cc} -i & \ 0 \\ 0 & i \\ \end{array} \right ) \left( \begin{array}{cc} \frac{i}{2} & \ \frac{1}{2} \\ \frac{-i}{2} & \frac{1}{2} \\ \end{array} \right )$$

We can now calculate the exponential of $A$ using

$$\exp(A)=S \cdot \exp(J) \cdot S^{-1}$$

Regards

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Thank you for accepting my answer, but I actually liked Thomas Andrews' answer myself as it solves the general case. Regards –  Amzoti Feb 5 '13 at 15:56
    
Nicely done; and what a good sport: having been accepted, but deferring to another solution!! –  amWhy May 4 '13 at 0:30
    
@amWhy: I always believe that the best answers win, although sometimes things like rep get in the way of that! Regards –  Amzoti May 4 '13 at 0:36
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Note that matrices of the form $xI+yA$ with real $x$ and $y$ behave exactly like complex numbers $x+yi$ under matrix addition and matrix multiplication, as well as topologically. Since additions, multiplications, and limits is all you need to define the exponential function (with the same definition in both cases), the answer is the same as $e^i$, translated back to a matrix by the correspondence $x+yi\mapsto xI+yA$.

Or in other words, $$\exp(A) = \begin{pmatrix} \cos(1) & -\sin(1) \\ \sin(1) & \cos(1)\end{pmatrix}$$

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