I have to chime in here because there are some who have posted here who obviously have never seen their child hysterically crying and actually scratching their skin off due to eczema. To suggest that a multi vitamin can help cure eczema is ludicrous.
Eczema is something I have dealt with since my son was born (he is now 9). He was diagnosed at 6 weeks old at Chicago's Children's Memorial with severe eczema over 100% of his body; he learned to rub his head, hands, arms, feet and legs against his crib sheet and clothing at 1 month old - to the point where his skin was rubbed off (BTW - he could do this in a matter of one or two minutes).
When a child has eczema, the possibility of infection is so high because of the open skin splits/cracks and sores from scratching. Bleach baths are a safe and effective way of controlling infections so that your child is not on antibiotics every single day of his life! Now, we are not talking a high concentration of bleach (about 1/4 cup to a full luke warm bath) - you are not pouring the bleach in while your child is in the tub, nor are you pouring it on your child! Also, these baths are only once a week and only as needed. Aveeno bath may work to a point, as will sea salt baths - they seem to help keep flares from happening once the eczema is under control. But when flares do occur, a bleach bath can work wonders. You need to stay ahead of eczema, don't wait for a flare to rear its ugly head, by then you are playing catch up and your child is the one who is suffering the consequence. We use Aquaphor lotion (in mass quantities) at least twice a day and after every hand washing. Never use soap - at the most, Dove for sensitive skin can be used (it's not soap) and never, ever use anything that contains alcohol (like waterless had sanitizer) which will instantly dry out the skin (not to mention burn, burn, burn!). My son does need to use topical medications as well to keep his condition under control. Allergens will certainly cause flares too.
Now, he has also been diagnosed with MC (with open skin splits, no matter how well bandaged, he seems to get every infection that comes down the pike). I have found that the occasional bleach bath keeps the secondary infection down so the use of topical antibiotics is not needed; the baths also help with the itching.
My best advice, is to educate yourself - read everything you can about it - keep an open mind and use common sense. Research a good PEDIATRIC dermatologist who specializes in eczema and take advise from her or him as well.