The answer is "yes." If you should die before your wife, the Social Security Administration will automatically raise her benefit to the amount you would have received. However, if she should pass away first, your amount will not be lowered.
The Social Security Administration's web site is actually quite user-friendly and has lots of information.
Unless someone is getting a spouse's benefit, it will NOT happen automatically, one has to file an application. Say a husband is getting $1000 per month from his work, and a wife is getting $600 from her work. That is more than one-half of his, so she would not be getting a spouse's benefit, just her own. But if the husband dies, she can file as a widow (and depending on age) could get up to $1000 per month. If she dies, husband would just continue to get his own benefit, and nothing more except for the $255 death benefit, which also requires an application. If the wife had a benefit of $400 on her own, she might be getting some more from his, and that would automatically convert, if she was above the full retirement age. Always call Social Security to report a death so they can stop the spouse's benefit and advise what you have to do to file, or tell you that is will convert automatically.