Mātangi is the name of one among the ten Mahāvidya. Specifically Mātangi is among the three Siddhividyas namely Kamalātmika, Mātangi and Bagalambe. Siddhividya refers to those who give the knowledge necessary to accomplish the native’s task in this world. They are said to be the quickest to attain results from and protect the native in the three lokas of Svarga (heaven/stars), Bhuva (sky/galaxy) and Bhu (earth) lokas respectively, i.e. the physically manifest lokas.
The remaining among the ten Mahāvidyas are said to be Vidyā whereas Kali and Tara are specifically Mahāvidyas, and most mantras and verses have been devoted to these two in the agamas and nigamas. This thus forms a hierarchy where Kamalātmika is the first to be worshiped by the sadhaka and Kali is the last.
Mātangi appeared in the north-western direction when the ten Mahāvidyas initially appeared in front of Shiva. This is the direction of vayu/air representing the strength and protection of 
every individual as air: 1) represents the prana or lifeforce in the body; 2) just as the earth has an atmosphere to protect us against the Suns rays, the body has a shield which protects it from outside circumstances. As a Mahavidya, Mātangi ensures perfect bodily health by giving good prana/lifeforce and gives freedom from disease as well. Astrologically this direction is presided over by the Moon. Whilst the Sun is said to represent Mātangi astrologically and thus ensures good health and spirituality when worshiped, Mātangi as a Mahāvidya reveals the weaknesses which the mind (represented by the moon) has been subject to, and the native overcomes astrological-doshas such as the solar-eclipse, nodes joined the Sun, etc.
Mātangi was born as a daughter from the Ṛṣi Mātanga. He was recognized as a Ṛṣi because the divine mother herself was born to him in the form of Mātangi. He was considered a nonconformist when it came to the idea of caste or varṇa. He was said to be a śudra himself and had no concern of the whole idea of the caste-system. When confronted by his student’s reg. this he illustrated his point by drawing a line in the ground. He then asked one person who belonged to a tribe of hills men to cross over this line. Each time the tribesman crossed over the line his caste had changed, as had his level of awareness and the knowledge he presented. Finally the tribesman reached the caste of a Brahmaṇa and presented a level of knowledge so exhaustive and refined that the students of the Ṛṣi could not compare.
For this reason Mātangi is regarded as the giver of ones varṇa or caste/occupation in the world. Astrologically the Sun represents the king, and as we learn from the traditions of eastern India, the king was responsible for deciding the occupation and thus also the caste of the kingdoms inhabitants. Similarly in the chart we analyze the fourth from the Sun to understand which karma is available for us to perform. Hence, the traditions of east India also regard the Sun as representing the Mātangi Mahāvidya in the charts.
Mātangi herself has about five or six distinct forms. These are: vaishya-matangi, sumukhi-matangi raja-matangi, karna-matangi, ucchiṣṭa chandali and matangi herself. Some believe that the form of Mātangi is worshiped depending on ones caste. Whilst this would seem applicable to maybe Raja-matangi and Vaishya-matangi, this cannot be so for ucchiṣṭa-chandali as it goes against the principle of the castes. I.e. a chandala is one who doesn’t conform to any of the castes, their regulations, rules, etc. and has rejected society in itself. Also most animals come under this grouping. That being so, a chandala would not comply to this if they were performing a mantra and sadhana for Mātangi, as they would have to follow the rules and regulations for performing Mahāvidya sadhana and thus begin to occupy a caste and occupation as a result.
Instead the specific Mahāvidya is worshipped depending on nature of the problems the native faces in life.
A story goes that Mātangi accepts the leftovers or ucchiṣṭa from the Gods, and for this reason she is known by the same name. However, some people also believe that she should be worshiped in the very same way, i.e. with leftovers, which is quite against the principles of worship adopted in ritual practices. It should be learnt that there are many forms of God which appear with the name ucchiṣṭa which literally means ‘having touched the lips’, i.e. ucchiṣṭa ganapati, ucchiṣṭa brahmanaspati, etc. These do not represent impure forms of God, but instead represent those forms from which you will speak their words! In case of Mātangi it is said that her worship will allow one to speak the words of Shiva himself which is of the highest knowledge.
For this reason Mātangi is often depicted with a parrot. In this form the parrot represents the student who is trying to mimic and copy the words of the guru, which are spoken by Mātangi.
Her words are that of the highest truth and she bestows the gift of unfailing speech and truthfulness to the sadhaka. She does this specifically in the forms of Mātangi, Karṇa-matangi and Ucchiṣṭa chandali, but generally this applies to every aspect of her form. This may be necessary in some contexts to take others out of untruth and lead people towards a greater ideal, and due to this she is associated with vaśikaraṇa or the ability to attract, persuade and control the minds of others. She takes on this role in her forms as Sumukhi-matangi, Raja-matangi and Vaishya-matangi. Especially Raja-matangi finds a particularly exhaustive depiction in the merutantra which finds traits of all forms of Matangi within her.
The reason Mātangi was depicted as a Chandala or outcaste is found in the story where Shiva took that same role. Shiva had decided to travel through the mountains without Parvati and he took on the appearance of a tribesman of the Matanga tribe. Parvati followed him in the same guise and Shiva, unaware of her real identity as his wife, became enticed and attracted to her disguise. This resulted in their conjugal union where after Parvati revealed her true identify to Shiva and made a vow to stay in the form that she had partaken. So she was known as Ucchiṣṭa Chandali for 1) having touched the lips of Shiva and 2) having taken the form of a Chandali. Because she was able to attract Shiva, she is associated with the act of vashikarana, and thus the worshipper of Mātangi developes the means to attract Shiva.
It was in this form that the couple established the Gṛṣneśvara Jyotirlinga in the Kanyaka forest, when by accident, whilst Mātangi was mixing sindoor and saffron with her thumb, a beam of light appeared in her palm. This then became known as Kumkumeśvara as it was made from kumkum, or Gṛṣneśvara as Parvati caused a lot of friction when rubbing the mixture with her thumb.
Method of selecting Mantra:
When a planet is debilitated in rasi, navamsa or even the shastyamsa-trimsamsa varga (d30) charts the mahavidya indicating the same is worshiped to overcome the punishment from previous lifetimes of performing adharma. For the Sun its debility indicates treating the patriachs wrongly, and disrespecting the king, government and similar officials. For this we advise the worship of Matangi. This should especially be done towards the period of Leo Narayana dasa.
Similarly when the Sun is eclipsed by the nodes we advise the Matangi worship to overcome the eclipse of the soul itself, and to hear the words of guidance from Guru.
We can also advise Matangi worship or Savitur Gayatri mantra when the Sun causes the combustion of other planets, or joins them in the navamsa. In such cases the native will completely ignore or find little happiness from the planets who are combust by the Sun.
The specific form of Matangi is then left to the intelligent astrologers to decide, but when in doubt the advise is to prescribe Prasiddha mantras for Mātangi.