Elysian families are neither inherently patriarchal or matriarchal, but tend to place more importance on social standing and age. For example, should your family be so blessed as to have a Paragon in it's ranks, they would almost certainly be the head of the family, regardless of their age. More commonly, however, it is simply successful older members of the family who will take charge and lead the family.
There is also a strong sense of fealty to family among Elysians to their family, regardless of social class. They always feel that regardless of how their actions impact others, they should always reflect well on their family before all else.