na jāyate mriyate vā kadācin
nāyaḿ bhūtvā bhavitā vā na bhūyaḥ
ajo nityaḥ śāśvato ‘yaḿ purāṇo
na hanyate hanyamāne śarīre

Translation of Bhagavad Gita 2.20

For the soul there is neither birth nor death at any time. He has not come into being, does not come into being, and will not come into being. He is unborn, eternal, ever-existing and primeval. He is not slain when the body is slain.

Commentary by Sri A.C. Bhaktivedanta Swami Prabhupada of Gaudiya Sampradaya:

Qualitatively, the small atomic fragmental part of the Supreme Spirit is one with the Supreme. He undergoes no changes like the body. Sometimes the soul is called the steady, or kuta-stha. The body is subject to six kinds of transformations. It takes its birth from the womb of the mother’s body, remains for some time, grows, produces some effects, gradually dwindles, and at last vanishes into oblivion. The soul, however, does not go through such changes. The soul is not born, but, because he takes on a material body, the body takes its birth. The soul does not take birth there, and the soul does not die. Anything which has birth also has death. And because the soul has no birth, he therefore has no past, present or future. He is eternal, ever-existing, and primeval—that is, there is no trace in history of his coming into being. Under the impression of the body, we seek the history of birth, etc., of the soul. The soul does not at any time become old, as the body does. The so-called old man, therefore, feels himself to be in the same spirit as in his childhood or youth. The changes of the body do not affect the soul. The soul does not deteriorate like a tree, nor anything material. The soul has no by-product either. The by-products of the body, namely children, are also different individual souls; and, owing to the body, they appear as children of a particular man. The body develops because of the soul’s presence, but the soul has neither offshoots nor change. Therefore, the soul is free from the six changes of the body.

In the Katha Upanishad (1.2.18) we also find a similar passage, which reads:

na jayate mriyate va vipascin
nayam kutascin na babhuva kascit
ajo nityah sasvato ’yam purano
na hanyate hanyamane sarire

[Bg. 2.20]

The meaning and purport of this verse is the same as in the Bhagavad-gita, but here in this verse there is one special word, vipascit, which means learned or with knowledge.

The soul is full of knowledge, or full always with consciousness. Therefore, consciousness is the symptom of the soul. Even if one does not find the soul within the heart, where he is situated, one can still understand the presence of the soul simply by the presence of consciousness. Sometimes we do not find the sun in the sky owing to clouds, or for some other reason, but the light of the sun is always there, and we are convinced that it is therefore daytime. As soon as there is a little light in the sky early in the morning, we can understand that the sun is in the sky. Similarly, since there is some consciousness in all bodies—whether man or animal—we can understand the presence of the soul. This consciousness of the soul is, however, different from the consciousness of the Supreme because the supreme consciousness is all-knowledge—past, present and future. The consciousness of the individual soul is prone to be forgetful. When he is forgetful of his real nature, he obtains education and enlightenment from the superior lessons of Krishna. But Krishna is not like the forgetful soul. If so, Krishna’s teachings of Bhagavad-gita would be useless.

There are two kinds of souls—namely the minute particle soul (anu-atma) and the Supersoul (vibhu-atma). This is also confirmed in the Katha Upanishad (1.2.20) in this way:

anor aniyan mahato mahiyan
atmasya jantor nihito guhayam
tam akratuh pasyati vita-soko
dhatuh prasadan mahimanam atmanah

“Both the Supersoul [Paramatma] and the atomic soul [jivatma] are situated on the same tree of the body within the same heart of the living being, and only one who has become free from all material desires as well as lamentations can, by the grace of the Supreme, understand the glories of the soul.” Krishna is the fountainhead of the Supersoul also, as it will be disclosed in the following chapters, and Arjuna is the atomic soul, forgetful of his real nature; therefore he requires to be enlightened by Krishna, or by His bona fide representative (the spiritual master).Commentary by Sri Vishvanatha Chakravarthi Thakur of Gaudiya Sampradaya:

The eternal nature of the jiva is here clearly proven. The first line negates the possibility of birth and death for the jiva at the present time. The second line negates birth and death in the past and the future. Therefore it is unborn (aja): in past, present and future. Because of no birth, it has no previous non-existence (prag abhava). It exists at all times (sasvatah), meaning that at no time in past, present or future, will it be destroyed (dhvamsa). Therefore it is eternal (nityah).

“But because the soul exists for a long time, it can grow old.”

“No, though it is old (pura), it is as if new (na for nava), because of absence of the six states of transformations.”

“With the death of the body, will it not die along with the body?”

“It does not die when the body dies. Because it does not have a relationship with the body, the soul is not subservient.”

Commentary by Sri Ramanuja of Sri Sampradaya:

2.20 As the self is eternal for the reasons mentioned (above), and hence free from modifications, it is said that all the attributes of the insentient (body) like birth, death etc., never touch the self. In this connection, as the statement, ‘It is never born, It never dies’ is in the present tense, it should be understood that the birth and death which are experienced by all in all bodies, do not touch the self. The statement ‘Having come into being once, It never ceases to be’ means that this self, having emerged at the beginning of a Kalpa (one aeon of manifestation) will not cease to be at the end of the Kalpa (i.e., will emerge again at the beginning of the next Kalpa unless It is liberated). This is the meaning — that birth at the beginning of a Kalpa in bodies such as those of Brahman and others, and death at the end of a Kalpa as stated in the scriptures, do not touch the self. Hence, the selves in all bodies, are unborn, and therefore eternal. It is abiding, not connected, like matter, with invisible modifications taking place. It is primeval; the meaning is that It existed from time immemorial; It is even new i.e., It is capable of being experienced always as fresh. Therefore, when the body is slain the self is not slain.
Commentary by Sri Sridhara Swami of Rudra Sampradaya:

The understanding that the eternal soul is never slain is being confirmed by its freedom from the six changes of material existence being: birth, existence, growth, modification, decay and death which are controlling all living entities high and low in the material existence. The Supreme Lord by declaring the eternal soul is not born confirms the absence of any birth and by declaring the eternal soul does not die confirms the absence of any destruction. The word va occurring twice is used in the sense of and as in and the immortal soul does not come into existence by birth. The immortal soul is already by nature itself eternal. So existence by conception is proven false due to the fact the eternal soul is birthless. That which is conceived attains existence after birth; but that which already eternally exists by its own nature cannot be conceived again. This is the purport of this verse. Constant means not subject to modification. This confirms the absence of growth. Eternal confirms the absence of decay. Ancient confirms the absence of ageing ever beyond time the eternal soul is ageless. Thus there is never a repetition regarding the existence of the eternal soul and the six changes of the physical of birth, existence, growth, modification, decay and death mentioned by Yaksa and other expounders of the Vedic scriptures as to the nature of things in this world never is applicable to the eternal soul which factually every living entity born from a womb possesses. The imperishable nature of the soul and the fact that the six changes of the physical body have no influence on it which is the topic presently under discussion has now been satisfactorily concluded and so it is confirmed that the eternal soul does not perish when the body perishes.
Commentary by Sri Madhvacharya of Brahma Sampradaya:

Here to emphasise the authority of the Vedic scriptures the Supreme Lord Krishna uses the words na jayate meaning never takes birth. The knowledge of the Lord is not born nor does it cease to be in the literal sense of the words. In all situations, time and places as well as in all stages it is never subject to modification. Nor is the knowledge of the soul ever subject to destruction and this is fully established in the Vedic scriptures. How one may ask? Due to the attributes like ajah the unborn referring to the Supreme Lord, sasvatah means permanent signifying that He remains in His spiritual form of consciousness and bliss without any modification, puranam means that the eternal embodied soul acquires another pura or body. Thus it can be understood that the soul is not ever destroyed even when the physical body perishes.

Now begins the summation.

The living entity and the Supreme Lord are both irrevocably established as unborn, eternal and indestructible. The Supreme Lord possessing a spiritual body does not come into existence by being associated with the material manifestation; but is eternally existing independently. Death correctly comprehended is merely the separation of the embodied soul from the physical body leaving it lifeless. Since for the Supreme Lord there is never any destruction of His form, there cannot be any connection to death by Him.

The eternal soul is also unborn and indestructible, otherwise creation could be considered simply a repetitive exercise going on and on absent of purpose. The eternal soul is also permanent but the eternal soul never exists independently; it has limited power, limited knowledge, incomplete in itself, dependent upon the transcendental energy of the Supreme Lord. In juxtaposition to that are the sublime attributes of the Supreme Lord who is omnipotent, omniscient and omnipresent. These are the natural attributes of the 2 there is no modifications to be found in them. Therefore the holy sages and rishi’s designated them both as sasvatah or permanent. Thus in the Visnu Purana because the living entity is embodied in countless bodies from the beginning of time it is known as puranah meaning ancient.
Commentary by Sri Keshava Kashmiri of Kumara Sampradaya:

As to questions of how the soul is neither the instigator of any actions or the recepient of any actions, Lord Krishna explains that the soul being eternal is not subject to the six modifications of material existence which are: birth, being, growth, transformation, dimunition and death. The soul is never born and never dies at any time. How can this be? It is not that the soul did not exist before and will not exist in the future. The soul exists eternally but opting the experiences of the material existence accepts a physical body. What did not exist in the past but is manifested only in the present is called conceived and that which exists now but will cease to exist eventually is called dead. But the soul is not conceived as it eternally exists and it will never die because it is immortal. With the 2 words ajah meaning unborn and nityah meaning eternal, it is made clear that the 2 modifications of birth and death are not applicable in regards to the soul. In order to also negate the idea of the soul posessing any modifications by even being, Lord Krishna explains that the soul is everlasting which means it is free from being or unbeing. The modification of any growth of the soul is neutralisd by the word puranah meaning ancient, that it always existed as it is. One posessing a body grows by the increase of the bodily parts; but the soul being bodiless does not increase it is the same eternally. The remaining 2 modifications of transformation and dimunition are rendered inapplicable by inference in understanding that the soul being eternal without any need to increase remains as it is and is changeless. Hence the soul being bereft of any of the six modifications of the physical body can never be destroyed even when the physical body is destroyed.
Commentary by Sri Adi Shankaracharya of Advaita Sampradaya:

2.20 Na kadacit, neverl; is ayam, this One; jayate, born i.e. the Self has no change in the form of being born — to which matter is subject –; va, and (– va is used in the sense of and); na mriyate, It never dies. By this is denied the final change in the form of destruction. The word (na) kadacit), never, is connected with the denial of all kinds of changes thus — never, is It born never does It die, and so on. Since ayam, this Self; bhutva, having come to exist, having experienced the process of origination; na, will not; bhuyah, again; abhavita, cease to be thereafter, therefore It does not die. For, in common parlance, that which ceases to exist after coming into being is said to die. From the use of the word va, nor, and na, it is understood that, unlike the body, this Self does not again come into existence after having been non-existent. Therefore It is not born. For, the words, ‘It is born’, are used with regard to something which comes into existence after having been non-existent. The Self is not like this. Therfore It is not born. Since this is so, therefore It is ajah, birthless; and since It does not die, therefore It is nityah, eternal. Although all changes become negated by the denial of the first and the last kinds of changes, still changes occuring in the middle [For the six kinds of changes see note under verse 2.10.-Tr.] should be denied with their own respective terms by which they are implied. Therefore the text says sasvatah, undecaying,. so that all the changes, viz youth etc., which have not been mentioned may become negated. The change in the form of decay is denied by the word sasvata, that which lasts for ever. In Its own nature It does not decay because It is free from parts. And again, since it is without qualities, there is no degeneration owing to the decay of any quality. Change in the form of growth, which is opposed to decay, is also denied by the word puranah, ancient. A thing that grows by the addition of some parts is said to increase and is also said to be new. But this Self was fresh even in the past due to Its partlessness. Thus It is puranah, i.e. It does not grow. So also, na hanyate, It is puranah, i.e. It does not grow. So also, na hanyate, It is not killed, It does not get transformed; even when sarire, the body; hanyamane, is killed, transformed. The verb ‘to kill’ has to be understood here in the sense of transformation, so that a tautology [This verse has already mentioned ‘death’ in the first line. If the verb han, to kill, is also taken in the sense of killing, then a tautology is unavoidable.-Tr.] may be avoided. In this mantra the six kinds of transformations, the material changes seen in the world, are denied in the Self. The meaning of the sentence is that the Self is devoid of all kinds of changes. Since this is so, therefore ‘both of them do not know’ — this is how the present mantra is connected to the earlier mantra.
Commentary by Sri Abhinavagupta of Kaula Tantra Sampradaya:    

2.20 Na jayate etc. Having not been at one time, This etc. : this Self will come to be, having not been at any time non-existent, but only having been existent. Therefore This is not born, This does not die too. For, having been [at one time], This will never be non-existent [at another time]; but certainly This will be [always] existent.
Sanskrit Shloka Without Transliteration Marks:

na jayate mriyate va kadacin
nayam bhutva bhavita va na bhuyah
ajo nityah sasvato ‘yam purano
na hanyate hanyamane sarire
Sanskrit to English Word for Word Meanings:

na — never; jāyate — takes birth; mriyate — dies; vā — either; kadācit — at any time (past, present or future); na — never; ayam — this; bhūtvā — having come into being; bhavitā — will come to be; vā — or; na — not; bhūyaḥ — or is again coming to be; ajaḥ — unborn; nityaḥ — eternal; śāśvataḥ — permanent; ayam — this; purāṇaḥ — the oldest; na — never; hanyate — is killed; hanyamāne — being killed; śarīre — the body.