Transverse Doppler Effect for a Moving Star

[Dazed&Confused]

Homework Statement

A star travels in a direction transverse to the line of observation from Earth, with a speed 0.5c. It also emits light with wavelength $\lambda_0$ in the rest frame of the star. Calculate the wavelength of the light as observed on Earth, and also the angle at which the light is emitted in the rest frame of the star. Comment briefly on how your result relates to relativistic time dilation.

Homework Equations

Lorentz Transformation.

$\textbf{p} = Ec$
$\frac{E}{c} = \frac{h}{\lambda_0}$

The Attempt at a Solution

Assuming the star is traveling in the positive $x$-axis and that it emits the photon in the positive $y$-axis, then the photon in the star's rest frame is

$$ \left ( \begin{array} \\
0 \\ \frac{h}{\lambda_0} \\ 0 \\ \frac{h}{\lambda_0} \end{array} \right ). $$

Multiplying by the Lorentz transformation matrix to the frame of the Earth from that of the star, we have

$$
\left ( \begin{array} \\ \gamma & 0 & 0 & \gamma \beta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \gamma \beta & 0 & 0 & \gamma \end{array} \right )
\left ( \begin{array} \\
0 \\ \frac{h}{\lambda_0} \\ 0 \\ \frac{h}{\lambda_0} \end{array} \right ) = \left ( \begin{array} \\ \gamma \beta \frac{h}{\lambda_0} \\ \frac{h}{\lambda_0} \\ 0 \\ \gamma \frac{h}{\lambda_0} \end{array} \right).
$$

So according to this the wavelength has decreased to $\frac{ \lambda_0}{\gamma}$, but that is different to what I've found on wikipedia, where it is the frequency that decreases.

What have I done wrong?

 

[Dazed&Confused]

Is it simply the wrong ordering of the transformation? Is the actual case that the light is in the $y$ direction in Earth's frame and so we multiply by the inverse matrix to find the photon in the stars frame ( which we already know the wavelength of)?