THE FOUR MATHAS:
After having chosen four disciples, Adi Shankaracharya gave the title of Shankaracharya to each one of them; he established four Shankaracharya Mathas in four corners of Bharat and made each of his disciples chief Acharya of each of the Mathas respectively. For the purpose of maintaining the unbroken tradition and prohibit any confusion from taking place about which are the original four Shankaracharya Mathas, Adi Shankaracharya wrote the Mathamnaya Setu which is a work dealing with the rules, description and other details of the four original Shankaracharya Mathas established by him. The Mathamnaya Setu verses 1-37 deals with the details of the four Shankaracharya Mathas stating their areas, names and other such information.
A) SHARADA MATHA: The Mathamnaya Setu verses 1-8 deals with the details of the Dwarka Sharada Matha. Sharada Matha has been stated to be the first Shankaracharya Matha situated in the western part of Bharat in Dwarka of the state of Gujrat. Realm of this Matha is Dwarka, denomination of its sanyasis is Keetwar which is defined as the sanyasi who always bestows mercy upon creatures and guides everyone on path of Dharma, it's natural glories (ranks of sanyasis) are Teerth and Ashram, presiding deities are stated to be Siddheswhar and Bhadrakali, first Acharya of this Matha is Vishwarup, sacred stream is Gomati, Gotra is Avigat, and Bramhchari is Swaroop the definition of which has been stated to be one who has knowledge of the self, carries out Dharma properly and and is always in bliss. This Matha has been given the responsibility of the Sama Veda and the associated Mahavakya is "Tat Tvam Asi" which is from the Chandogya Upanishad Chapter 6 Section 8 verse 7 which means that thou art. The sanyasis who are steadily dedicated to sanyas ashram and free from being subject to life and death is called Ashram and the sanyasi who takes a holy dip chanting the Mahavakya of this Matha in confluence of sacred rivers is called Teerth. The territories situated in the western direction bounded by Sindhu, Sauvir, Saurashtra and Maharashtra falls under the administration of this Matha.
B) GOVARDHAN MATHA: The Mathamnaya Setu verses 10-17 deals with the details of the Puri Govardhan Matha. Govardhan Matha has been stated to be the second Shankaracharya Matha situated in the eastern part of Bharat in Puri of the state of Odissa. Realm of this Matha is Puroshottam, denomination of its sanyasis is Bhogvar which is defined as the sanyasi who precludes humans from enjoyment of senses and such objects, it's natural glories (ranks of sanyasis) are Van and Aranya, presiding deities are stated to be Jagannath and Vimala, first Acharya of this Matha is Padmapad, sacred stream is Mahodadhi, Gotra is Kashyap, and Bramhchari is Prakash the definition of which has been stated to be one who is skilled in practice of yoga, has knowledge of the self, is luminous and enlightened. This Matha has been given the responsibility of the Rig Veda and the associated Mahavakya is "Prajnanam Bramh" which is from the Aitareya Upanishad Chapter 3 Section 1 verse 3 which means consciousness is Bramh. The sanyasi who roams in the forests without any shackles of hope or despair is called Van and the sanyasi who renouncing the world stays in the forest alone is called Aranya. The territories situated in the eastern direction bounded by Anga, Baga, Kalinga, Magadha, Utkal and Barbar falls under the administration of this Matha.
C) JYOTIR MATHA: The Mathamnaya Setu verses 18-27 deals with the details of the Chamoli Jyotir Matha. Jyotir Matha has been stated to be the third Shankaracharya Matha situated in the northern part of Bharat in Chamoli of the state of Uttarakhand. Realm of this Matha is Badrikashram, denomination of its sanyasis is Anandvar which is defined as the sanyasi who precludes humans from worldly pleasures, it's natural glories (ranks of sanyasis) are Giri, Parvat and Sagar, presiding deities are stated to be Narayan and Purnagiri, first Acharya of this Matha is Totak, sacred stream is Alokananda, Gotra is Bhrigu, and Bramhchari is Anand the definition of which has been stated to be one who knows the absolute truth who always thinks about knowledge and truth and infinite. This Matha has been given the responsibility of the Atharva Veda and the associated Mahavakya is "Ayamatma Bramh" which is from the Mandukya Upanishad Chapter 1 verse 2 which means the soul is Bramh. The sanyasi who roams in the forests and hills engaged in study of Gita, is sober and has firm prudence is called Giri, the sanyasi who renouncing the world stays in the mountain valley, who's knowledge is mature, and specifically quintessence is called Parvat and the sanyasi who picks the gem of knowledge from the deep sea of substance and never exceeds their limits is called Sagar. This Matha is also known as Shree Matha. The territories situated in the northern direction bounded by Kuru, Kashmir, Kamboj, Panchal and some other adjacent territories falls under the administration of this Matha.
D) SHRINGERI MATHA: The Mathamnaya Setu verses 28-37 deals with the details of the Shringeri Matha. Shringeri Matha has been stated to be the fourth Shankaracharya Matha situated in the southern part of Bharat in Shringeri of the state of Karnataka. Realm of this Matha is Ramesvar, denomination of its sanyasis is Bhurivar which is defined as the sanyasi who precludes humans from obsession of excessive material wealth, it's natural glories (ranks of sanyasis) are Swarasatti, Bharati and Puri, presiding deities are stated to be Adivarah and Kamakshi, first Acharya of this Matha is Hastamalak, sacred stream is Tungabhadra, Gotra is Bhur Bhuva, and Bramhchari is Chaitanya the definition of which has been stated to be the enlightened one who knows the universal spirit as pure, infinite, undecaying and un-perceiveable. This Matha has been given the responsibility of the Yajur Veda and the associated Mahavakya is "Aham Bramhasmi" which is from the Brihadaranyaka Upanishad Chapter 1 Section 4 verse 10 which means the I am Bramh. The sanyasi who is always engaged in knowledge of Shri Vidya is called Saraswatti, the sanyasi who is full of bulk of knowledge and unaffected by distress is called Bharati and the sanyasi who has knowledge of the supreme reality and is always engaged in samadhi is called Puri. The territories situated in the southern direction bounded by Andhra, Dravida, Karnataka, Kerala, and other southern regions falls under the administration of this Matha.
Ps: The Mathamnaya Setu does not mention the kanchi Matha nor does the Madhaviya Shankara Digvijaya as pointed in an earlier answer. However the kanchi Matha is recognized as a Dharma Kendra but its presiding Acharya is called Parmachary and not Shankaracharya, the authority of being Shankaracharya belongs to the chaturamnaya mathas alone and not to kanchi nor can it overrule the Shankaracharya mathas however the present Shankaracharyas are of the opinion that Adi Shankaracharya did establish the kanchi Matha (u can check their videos on YouTube on this topic).
JAI VEERBHADRA
JAI SHANKARACHARYA
Credits of the writing and compilation belongs to: Jalandhar.