In the Jewish Virtual Library, I came across this:
For tribal affiliation, priestly status and royalty, patrilineal descent determines membership.
I'm specifically interested in the aspect of royalty. Where (either in the Tanakh or the Oral Torah) is this outlined? Is rationale given in that particular source? If not, what rationale is commonly offered, if any?
The halacha is that there cannot be a female king. Rambam brings this halacha in Melachim uMilchamot 1:5
אין מעמידין אשה במלכות שנאמר עליך מלך ולא מלכה.
We do not raise a woman to kingship, as it says "over you a king" and not a queen.
(Translation mine)
The verse that he is citing is Devarim 17:15 which says that Israel may appoint a king. He makes a diyuk in the language "king" to the exclusion of a queen.
Since the ruler must be male, it wouldn't make much sense for royalty to be passed matrilineally.
The Rambam brings the Midrash Sifri on Deuteronomy (17, 16) "שׂוֹם תָּשִׂים עָלֶיךָ מֶלֶךְ" that says: "A king and not a queen", and writes that a woman cannot be appointed to be "king".
Regarding the heritage rules:
When a king dies, his son will rule under him (see Gemara, Kritut, 5), as outlined in Deuteronomy (17, 20):
לְמַעַן יַאֲרִיךְ יָמִים עַל מַמְלַכְתּוֹ הוּא וּבָנָיו בְּקֶרֶב יִשְׂרָאֵל
If he has several sons, the oldest will succeed. If his son is a Katan, the throne "waits" for him to come of age. If he has no sons, the same rules that apply to heritage priority detailed in Bamidbar (chapter 27) will be considered.
A succedding son must be Yerei Shamaim (observant) in order to be worthy of taking over the throne.
