The senior most direct disciples of Sri Ananda Tirtha (Madhwa) were Sri Padmanbha Tirtha, Sri Narahari Tirtha, Sri Madhava Tirtha and Sri Akshobhya Tirtha. In the same order they were the pontiffs of what latter became known as the Sri Uttaradi Mutt with an ending year of 1324 AD, 1332 AD, 1350 AD and 1365 AD, respectively. Some suggest that Sri Akshobhya Tirtha obtained his sannyasa ashrama from Sri Madhava Tirtha, however from my understanding Akshobhya Tirtha was a sannyasi when Madhwa installed him with the others at Kanva tirtha to be the heads of the various Mutts
As aforementioned by Sri Banejee Govindacaryaji, Sri Madhava Tirtha also established the Majjige Halli Mutt near Mulubagal (Karnataka) and deputed Sri Madhurai Tirtha (one of his desciples) to look after the mutt. The Majjjige Halli Mutt worships the Veera Rama Idol personally worshipped by Sri Madhava Tirtha. He also started Kanva Mutt.
He also gave 'ashrama' to Sri Akshobhya Tirtha and installed him as the Pith Adipatya or pontiff of Sri Uttaradi Mutt to worship the idols/Deities which were personally worshipped by Srimad Acharya (Madhwa) himeself.
Once a marvellous incident happened when Sri Madhava Tirtha was engaged in the worship of Digvijaya Rama that Narahari brought back for Madhwa. All of a sudden he moved from his place and made prostration to some great personality visible to himself and invisible to others and sat with folded hands. The whole pooja was over and Sri Madhava Tirtha most reverently took the sacred tirtha (holy water) from one who was invisible. The whole incident was clarified afterwards by Sri Madhava Tirtha himself to satisfy the curiosity of the devout spectators. He said - "The Deity of Vamsha Rama which I have now acquired with Providential grace has been endowed with special sacredness of Srimad Acharya (Madhwa) who came in person to worship the Deity. He worshipped in his usual grand way, took tirtha (charan-amrita) himself and blessed me with tirtha afterwards". The whole gathering was astounded at the great event. A similar incident was also reported to have been experienced by Sri Akshobya Tirtha when he was worshiiping Sri Prasanna Vittala in Pandarapur..
Sripad Madhava Tirtha was quite often confused with, or known as 'the Madhwacarya' or Madhwa, due to the similarity in name.
There is also a 'mayavadin' of the same name of almost the same time,
consequently these two are often confused. As B.N.K.Sharma the Madhwa scholar
has pointed out in his book (B.N.K. Sharma. 1961. "History of the Dvaita
school of Vedanta", Page 229.). The story refered to in this connection
of how the other Madhava Tirtha founded the city of Vijayanagar after the
discovery of a huge amount of hidden treasure is as follows. Before this,
however, Madhava Tirtha lived an austere life as an ascetic in the mountains
of Karnataka, who once found himself being frequented by a shepherd of
the name Bukka. This poor shepherd had heard that Madhava Tirtha, the great
'sannyasi', was there absorbed in thoughts of Lord Narayana, and so he
decided to daily bring him some simple foodstuffs. Madhava Tirtha blessed
him saying, "One day you shall be the King of all Industan." By this blessing,
immediately all the local shepherds made him their head, and he became
known as the King of that local country which governed five groups of communities
- Canara, Taligas, Canguivaro, Negapatao and Badagas. In this kingdom he
became known as Boca Rao and reigned for thirty seven years by the blessings
of the great Madhava Tirtha. Bukka conquered many kingdoms from the time
he came to rule in 1343 A.D.
In his usual insistant manner K. Raghupathi Rao says, "The name Madhava Tirtha referred to by you refers to Vidyaranya, the 'mayavadin guru', who founded the Vijayanagar kingdom through Bukka. He was at Sringeri too. He was defeated in debate by Akshobhya Tirtha, the debate having been umpired by the great Sri Vaishnava Sri Vedanta Desika."(K.Raghupathi Rao. 12th January 1991. Letter to JTCd.)
The other Madhava Tirtha (Vidyaranya) was the son of Mayana and Srimati and the elder brother of Sayana, and at one time according to some was a minister of King Bukka of Vijayanagar.
For some reason best known to Tikacarya Jaya Tirtha, while commenting on almost every other 'acarya' completely ignored any works of Madhava Tirtha, not even mentioning him in any way shape or form!(BNK Sharma. 1961. History of the Dvaita School of Vedanta. page 229.) Some Madhwas that I spoke to even suggested that Madhava Tirtha left the association of Vaishnavas, going over to the 'enemy camp'. Other Madhwas' were extremely upset to think that anyone could even suggest such a thing. The Poona Prajna Vidya Pitha said, "However, Sri Madhava Tirtha, the disciple of Sri Madhvacharya is not related in the establishment of Vijayanagara."(Tirumala Kulkarni, Bangalore).
BNK Sharma suggests that it was simply that, "......he had probably no literary leanings."(BNK Sharma. 1961. History of the Dvaita School of Vedanta. page 229.) And that this is why there is next to nothing is writen in the 'mutt' records to glorify his deeds. This cannot be said for sure, but one would think that as the pontiff for 17 years something would be known.
All that we could find is that Sripad Madhava Tirtha was in office at the Vedanta Pitha from 1333 A.D., until 1350 A.D., when he passed away. Some say that he made a commentary on the Parasara Smrti called Parasara Madhwa-vijaya, and some other books have also been accredited to him (Rg, Yajus and Saama Veda commentaries), most people consider this very complimentary, but possibly unfounded. As little is known of him, there is much confusion maybe mis-identifying him with others.
It is said that his bodily remains were entombed at Hampi, but have since been moved to Manur on the Bhima River of Bijapur District. Very recently, a mritika brindavana of Sri Madhava Tirtha was installed in Mulubagal. He founded a mutt of his own near there under the name of Majjigenahalli Matha just near by Mulbagal and deputed Sri Madhurai Tirtha (one of his desciples) to look after it. The Majjjige Halli Mutt worships the Deity of Veera Rama personally worshipped by Sri Madhava Tirtha. He is also reputed by some to have started Kanva Mutt..
The Mahalaya Amavasya (which usually falls in the month of Aswin) is
this great saint's aradhana (disappearance anirversary).