You would do it by hand with a little bit of javascript.
It sounds like you want to show an image associated with a rollover of the name... most likely you would add an onmouseover="" javascript link to the <a> tag that you have linked and have code that updates the image src.
The code might look like:
<a href="yourpage.php?artist=<?php echo($row_rs['artist_id']); ?>" onmouseover="document.getElementById('imageid').src = '<?php echo($row_rs['artist_image_location]); ?>'"><?php echo($row_rs['artist_name']); ?></a>