Help with Matrices and Eigenvalues

Firstly sorry if this is in the wrong section, I wasn't sure where to put it, please feel free to move it.

I was just looking to get some help on a question I'm stuck on.

Vector r =

x

y

z

Rotation R =

cos(θ) 0 sin(θ)

0 1 0

-sin(θ) 0 cos(θ)

r' = Rr

It asks me to prove that

r'.r' = r.r

I found r' to be

xcos(θ) + xsinθ

y

-zsin(θ) + xcos(θ)

To get r'.r' am I right to just multiply two of the above together, as in

(xcos(θ) + xsin(θ))(xcos(θ) + xsin(θ))

(yy)

(-zsin(θ) + xcos(θ))(-zsin(θ) + xcos(θ))

Because this is the way I did it and it doesn't lead to the same answer.

Obviously these should all have big brackets around them, but I am unsure of how to represent them on here, if someone would advise me I would gladly fix that.

Second part of the question is about eigenvalues, it asks me to find the three eigenvalues of R.

I used the formula

det(m - λI)

Where m = R, λ = the eigenvalues and I is the appropriate identity matrix

I end up with λ = cos() or 1, which is clearly wrong as there isn't enough answers.

Somebody in class mentioned imaginary numbers, but I'm unsure as to how to proceed.

Any advice with either part would be much appreciated, I can provide working out of what I have done so far if wanted/needed.

Thank you.

Re: Help with Matrices and Eigenvalues

You're not doing it well, revise in your textbook how to multiply two matrices:

$\displaystyle r'=\begin{bmatrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \end{bmatrix} \begin{bmatrix}x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x\cos(\theta)+z\sin(\theta) \\ y \\ -x\sin(\theta)+z\cos(\theta) \end{bmatrix}$

To multiply two vectors you're supposed to use the definition of dot product:

$\displaystyle \begin{bmatrix}a_1 \\ b_1 \\ c_1 \end{bmatrix} \begin{bmatrix}a_2 \\ b_2 \\ c_2 \end{bmatrix}=a_1a_2+b_1b_2+c_1c_2$,

with this information try to check $\displaystyle r'\cdot r' =r\cdot r$.

For the second part a possible way is to use cofactors along the second row, so that you get

$\displaystyle \det\left(\begin{bmatrix} \cos(\theta)-\lambda & 0 & \sin(\theta) \\ 0 & 1-\lambda & 0 \\ -\sin(\theta) & 0 & \cos(\theta)-\lambda \end{bmatrix}\right)=(1-\lambda)(\lambda^2-2\lambda \cos(\theta)+1)$

and this is 0 if $\displaystyle \lambda=1$ or $\displaystyle \lambda^2-2\lambda \cos(\theta)+1=0$ (use the quadratic formula).