Yes - this is a Brown Recluse. The fiddle is obvious and you have a very good picture of its eyes...it appears to have 3 sets of 2 eyes - which is exactly what defines a BR.
My family discovered we had them, when my daughter was bitten by one in May - we know for sure because she captured the spider that bit her and we brought it to the Dr.
Look up the "how to's" on a few websites about dealing with them...here are a few biggies-
1- Set glue traps - tons of them - This post may get deleted for this, but Walmart has good ones for very cheap - put at least 3 traps per normal room (against the walls), a few under each bed, each closet, behind/under couches water heaters, washer dryers etc...don't skimp on these...catch all that you can- my philosophy now is that I would rather se them in the trap than anywhere else in the house. Check them each morning so you know the new hot spots.
2- Seal the entire outside of your house...gotta do it...every crack, every window sill, every where...except of course the foundation spaces...
3- clean and vac every wall, corner, etc...you may find more in the traps the next morning after you do this, but that is good...you are forcing those buggers out of hiding.
4- Pull all beds away from walls (2 inches or so) and do not let bed skirts touch the floor...this will help prevent those guys from getting into your bed and help you sleep better while you deal with this. If you want to (we did this) put a trap next to each bed post for added protection and to help you sleep better at night.
5- Make sure your exterminator (get one) knows BR spiders...
6- This is the toughest one...stay calm...the odds of being bitten are small...but teach everyone to be a bit more carefull...check clothing before putting it on, do not leave clothing on the floor, and keep the house a bit cleaner than normal.
What state do you live in???
Good luck...I feel your anxiety now...