Quick way of raising matrices to indicies

[thomas49th]

Homework Statement

A given matrix <br /> A(x) =\left | \begin{array}{cc} cos(x)&amp;sin(x)\\ -sin(x)&amp;cos(x) \end{array}\right | <br />

Find the term in the first row and second column A^{5}(x) when x = 18°

Homework Equations

The Attempt at a Solution

Instead of multiplying the matrix again and again by itself in a iterative procedure is there any quicker way?

Thanks
Thomas

 

[Ben Niehoff]

For this particular matrix, why don't you find A^2 for general x, and use your knowledge of trigonometric identities...then you may see a pattern.

 

[thomas49th]

A^2 = [ cos2x 2cosxsinx
-2cosxsinx cos2x ]

what I want is cos²x + sin²x so i get one, but what I got at the mooment doesn't seem to get me any where :( I wanted somthing like if the power is odd then you get -1 and even 1 in the first element, but can't see that happening :(

 

[Billy Bob]

Since 2 cos x sin x = sin 2x, a pattern is emerging, just not the one you thought.

 

[thomas49th]

aha so the formula for that specific elemnt is \sin 2^{n-1}x

Correct?

Thanks
Tom

 

[Billy Bob]

Maybe for A^{2^{n-1}} but that doesn't help with A^5.

In other words, multiply A^2 by A, not by A^2, to find A^3.

(I know you know this -- your brain just went into overdrive and caused a blunder.)

 

[thomas49th]

ahhh I am getting
A^3
<br /> <br /> A(x) =\left | \begin{array}{cc} cos(x)cos(2x)-sin(x)sin(2x)&amp; cos(x)cos(2x) +sin(x)sin(2x)\\ -cos(x)cos(2x)-sin(x)sin(2x)&amp; cos(x)cos(2x)-sin(x)sin(2x) \end{array}\right | <br /> <br />

where is this pattern? I can see they're all similar
I might as well just expand to A^5 now :(

 

[Billy Bob]

What is the trig identity for cos(A+B)? For sin(A+B)?

Also, your A^3 has two incorrect entries.

 

[thomas49th]

cos(A+B) = cosACosB - sinAsinB
sin(A+B) = sinAcosB + cosAsinB

right?

I can see how we can use RCos(x+a) for A^3 but I want to get A^5. There must be some relationship!

Which of the terms are incorrect in A^3btw for A^2 i got [cos2x sin2x
-sin2x cos2x]

Thanks
Thomas

 

[Borek]

I am sure BB means something much faster, but whenever I hear about anything raised to large integer power I think in terms of powers of 2, in this case it means A⁵=A*(A²)² - that means just three multiplications instead of 5. The larger the exponent, the more you save.

 

[Billy Bob]

btw for A^2 i got [cos2x sin2x
-sin2x cos2x]

Can't you guess the correct pattern just from this!

Additional hint: A(x) corresponds to rotation through an angle of x.

 

[thomas49th]

I can see A^2 = sin2x

but I cannot see why that is so argghh! must be something really simple!

 

[Billy Bob]

A=
[cos x sin x
-sin x cos x]

A^2=
[cos 2x sin 2x
-sin 2x cos 2x]

A^3=
what would you guess! Yes, it's very simple, sorry.

Now compute A^3, and if it doesn't match your guess, then your computation is probably wrong, not your guess!

 

[HallsofIvy]

You might also note that
\begin{bmatrix}cos(\theta) &amp; sin(\theta) \\ -sin(\theta) &amp; sin(\theta)\end{bmatrix}
corresponds to a rotation by -x degrees. Multiplying it by itself 5 times is the same as rotating by -5x degrees. In this case that is -5(18)= -90 degrees. The result is suprisingly simple.

 

[thomas49th]

yeah I didn't think of that!

still need to expand it manually to see the maths but bed time now

Thanks for the help!
Thomas