The Sri vaishavaites of some groupings follow the Swayam Acharya Sampradaya.
I have a close friend who is a Swayam Acharya.
Unfortunately, though he is a devout Iyengar,he was unable to say what it was excepting that he was following the Acharya of a Mutt.
When I checked with some of m other Iyengar friends, they also did not have the details.
I have referred some texts, the internet and am sharing the information.
Suggestions for improvement is welcome.
The earliest name the Puranas ascribe to those who worship Lord Vishnu is ‘Bhagavatha’
Later it was meant to include those who follow Pancharatra.
Please read my post on Pancharatra.
Bhagavatha means one who sings. praises the Glory of Bhagwan, here it refers only to Vishnu,
Technically this is incorrect as it refers any one who praises Bhagawan,
Please read my post on Bhagwan, God in Hinduism.
Equally incorrect to say that Ishwara refers only to Lord Shiva.
Iswara means personal God , that’s all.
Now to Swayam Acharya.
Later when many Devathas were being worshiped, to distinguish those who worship Vishnu, the term Vaishnava was coined.
As Vishnu is to be worshiped always along with Lakshmi, Sri, has been added.
Thus was born the term Sri Vaishnava.
Swayam Acharya means one who is an Acharya to himself.
In the earlier days, when Brahmins were following all the duties of a Brahmin, including Agnihotra, the tradition was handed over to the son.
Thus the father becomes the first Acharya.
In Upanayana, the Gayathri is first taught by the father.
In all cases Father is the first Acharya.
Sruthi says Pitru devo Bhava, Acharya Devo Bhava in this context.
“Swayam Acharya sampradayam:
In olden days, all the Vaishvaites were very orthodox, performing daily Agni Hothram in their houses. Therefore the father of the family himself was the Acharyan for the members of that family. He was initiating the members into Sri Vaishnava sampradayam by Samasrayanam process.
Matams and Munitrayam sampradayam:
Now a days, most of the family heads do not perform agni hotrams and follow the sampradayam very strictly. So for the purpose of Samasrayanam, we approach Matams. The matadhipathi/Swami/Acharya initiates us into sampradayam.
This term is used for Sri Vaishnavas who are descendants of the original 74 disciples of Ramanuja, and who continue to have association with learned members of their extended family who administer the panca-samskAra (samASrayaNam) and minister to their needs. Many of our most learned acharyas came from "svayam-AchArya" families -- Desika, Pillai Lokacharya, etc. This is because the concept of a maTha and therefore a sampradAya based on the maTha dates from the 14th century, much after Sri Ramanuja's time. Some notable svayam-AchArya families are Prativaadi Bhayankaram, Tatacharya, Tirumalai Anandaanpillai, Nallaan Chakravarti, etc. There are many scholars among these families even today, as they have a strong sense of connection to the tradition and maintain a certain level of scholarship. Not all descendants of the 74 original disciples are still considered "svayam AchArya". At some point, some of the descendants drifted away from their traditional acharya or did not have a strong scholar in their family, and consequently became associated with some maTham or other swami. So there will be Sri Vaishnavas who bear the appellation "Kidambi", "Vangipuram", "Tatacharya", etc., who at one point belonged to svayam AchArya purusha families, but who now follow some other swami. It should be pointed out that some people object to the term "svayam AchArya purusha" (not the concept behind it) with the feeling that only one person is truly a "svayam AchArya", an acharya for himself -- the Lord.