Who is the goddess/matrika kaumari, and how is she worshipped?

 The Matrika goddess Kaumari (aka Kumari, Kartiki, Kartikeyani, Karrtikkeyi, Ambika, sometimes called Lumadhi Ajima in Kathmandu) is the emanated and personified Shakti of Kumara (aka Kartikeya/Skanda/Murugan etc.). So she resembles Kumara but female, with his same peacock mount and banner, and the characteristic attitudes of valor and courage. Like him, sometimes she appears with six heads. Her skin is yellow and her clothes are red. She is a portion of Parashakti, and her relation to Kumara is motherly as well as partly his own self. She is the consort of Chanda Bhairava. She has four or twelve arms. In her four-armed form she usually holds a Shakti spear and a rooster, and performs Abhaya and Varada Mudras. In her twelve-armed form she often holds a Shakti spear, a banner showing a peacock, a rod, a bow and arrow, a bell, a lotus, a bowl, and an axe. Often she holds silver coins. She wears a karandamukuta (cylindrical crown). Sometimes she has a child on her lap, though notably she is always very young, never older than sixteen years old in appearance.

Kaumari, like the other Matrika goddesses, is a wild and dangerous force of raw power, like an unshielded live wire - sheer energy, potent but just as likely to harm as help if not handled with expert care. She is a ferocious demon-slayer, a fire-breathing, death-spraying destructrix, an exultant blood-drinker, a wantonly bloodlusty indiscriminate devourer once she gets frenzied. She is particularly dangerous to unborn and infant children, whom she often kills with fevers; or even anyone under the age of sixteen. She can generally cause diseases, infertility (particularly in women), and other calamities. She is safer if worshipped together with Ganesha, Bhairava, Virabhadra, Narasingha, Durga, or of course Kumara himself - powerful warrior forms whom she respects. When worshipped properly she can grant potent protection, Bhukti (fulfillment of desires), and even Mukti. To clarify, it’s not that she defaults to harmful unless appeased; rather, she is unpredictable. Even without correct methods, you might get lucky. All of this applies to all the Matrika goddesses, including Kaumari but not specific to her.

As for how she is worshipped, again she is almost always worshipped together with other Matrika goddesses - at least four others, usually at least six others, sometimes more - rather than Kaumari alone. By far their most common worship, and the only way in which I myself have participated in their worship, is in the great Chandi Homa. At a certain point in the Chandi Homa, Kaumari is evoked with this shloka and given offerings:

“Kaumari shakti hasta cha mayura varavahana

yoddhumabhyayayau daityan Ambika Guharupini!”

And later, all the Matrikas are given offerings (traditionally a sprinkle of blood into the fire, in modern times often substituted with pomegranate juice) with this line:

“Tesham Matrigano jato nanartashrin madoddhatah!”

“The multitude of the Matrikas, exultant from drinking the demon’s blood, danced vigorously with joy!”

Her other main Mantra is:

“Mayurakukkutavrte Mahashakti dhare nadhe

Kaumari rupasansthane Narayani namostute”

As for their far rarer and more dedicated worship outside this context, the Matrikas are worshipped on the night of the dark moon, together with their ganas (spirit followers). They are traditionally worshipped in nocturnal Vamachara Tantrika rites, with ritual blood sacrifice and offerings of meat and wine. The specific methods are secretive, involving certain Yantras (largely circular in pattern), and sometimes also involve pain, such as deliberately burning oneself with the Arati lamp. They are worshipped in certain trees (an udumbara fig tree for Kaumari), in north-facing shrines near lakes or rivers, at crossroads, in caves (such as the Matrika caves at Udayagiri, Bhopal; four different Matrika caves at Ellora and Elephanta; and in the caves of Shamlaji, Gujarat, where Kaumari is especially emphasized), in crematory grounds, and in certain Tantrik temples - sometimes entire dedicated temples, but more often in Matrika shrines within larger temple complexes (in the northern part of the complex), or separate Matrika shrines in villages. In temples and caves the Matrikas are more often worshipped in full Murti forms; in smaller shrines, more often just as a circle of uncarven stones smeared with blood-red vermillion. In Tantrik rites done at home, they are usually evoked into images drawn with rice flour on plates, or simply into supari nuts, as it is usually considered too unsafe to keep their Murtis at home. In this context they are usually offered fruits, perfume, and and flowers (specific flowers given in the Devi Purana - a garland of red flowers for Kaumari) or a normal Panchopachara Puja, rather than risking offering flesh or blood and potentially riling them into a feeding frenzy. The full blood offerings are not done every day even at dedicated Matrika temples either; rather, the daily Bali is usually just of grain, with blood offerings on dark moon nights and especially during Navaratri.

Another important mode of Matrika worship is that they are worshipped as the sounds of the Sanskrit alphabet, the essential powers of Mantra Shastra, and are invoked as such into the Tantrika Sadhaka’s own body in the process of Matrika Nyasa. This is an advanced Tantrik rite which requires proper initiation. Kaumari is associated with the Manipura Chakra.

Specifically to prevent them from killing fetuses and newborns, Matrikas are worshipped by women and offered traditional bridal clothing and ornaments, to evoke their wifely, domestically feminine nature rather than their wild, feral predatory nature.

The main land of Matrika worship is Nepal, where Buddhists as well as Hindus worship them. The primary center of Matrika worship is specifically the Kathmandu valley. I myself have visited several Vamachara Tantrika temples there where the Matrikas, including Kaumari, are worshipped (though not during Matrika worship, which generally is not open to outsiders due to its dangerous nature, except during certain festivals). These included the Rakta Kali temple in Kathmandu, the Dakshinkali temple in the forest outside Pharping, and the Akasha Bhairava temple in Kathmandu, where with permission of the attendant priestess I took this photo of Kaumari (she is kept behind a pane of glass so that visitors can see but not touch the Murti, since it is alive with Prana Pratishtha and considered spiritually dangerous to touch without Tantrik expertise):

Notably, this is separate from the Pancha Kumaris of Kathmandu, who are different deities entirely.

The three ancient cities of the valley - Kathmandu, Bhaktapur, and Patan - each worship the Matrika goddesses as ashtadigpalakas or guardians of the eight directions (the four cardinal and four ordinal directions). Each of these cities had eight Matrika Pithas established in a circle (Mandala) around them, one in each direction. Kaumari is the guardian of the eastern direction. This was done in ancient times and the cities have expanded beyond these boundaries since then, so only the historical cores of the cities are within the three protective Matrika Mandalas. Each of the eight Pithas is dedicated to one particular Matrika (so one of them, in each of the three cities, is dedicated to Kaumari), with the other Matrikas secondarily worshipped there. These Matrikas are worshipped with blood sacrifice each year during the autumn Navaratri. Also, all of the smaller towns and villages in the valley have their own temples with brass Murtis of the Ashtamatrikas, and in the autumn Navaratri they bring them to the nearest of the three great cities in a festival parade and place each Matrika at her dedicated Pitha - so, each village’s brass Kaumari Murti is brought to the Kaumari Pitha in the city, and partakes in the blood sacrifice before being taken home after Navaratri.

There is an additional ritual associated with these Ashtamatrika Pithas of the valley, drawn from the Bhavani Upanishad, Varaha Purana, Vishnudharmottara Purana, and Tantraraja Tantra. Each Matrika is associated with a certain vice - anger, greed, etc. - and people perform a pilgrimage to all twenty-four Matrika Pithas of the three cities, worshipping each Matrika with Puja to gain control over their respective vices. Buddhists as well as Hindus perform this ritual.

Here is a photo I took of one of her street shrines in Kathmandu: