yasmāt kṣaram atīto ’ham
akṣarād api cottamaḥ
ato ’smi loke vede ca
prathitaḥ puruṣottamaḥ

Translation of Bhagavad Gita 15.18

Because I am transcendental, beyond both the fallible and the infallible, and because I am the greatest, I am celebrated both in the world and in the Vedas as that Supreme Person.

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

No one can surpass the Supreme Personality of Godhead, Krishna—neither the conditioned soul nor the liberated soul. He is therefore the greatest of personalities. Now it is clear here that the living entities and the Supreme Personality of Godhead are individuals. The difference is that the living entities, either in the conditioned state or in the liberated state, cannot surpass in quantity the inconceivable potencies of the Supreme Personality of Godhead. It is incorrect to think of the Supreme Lord and the living entities as being on the same level or equal in all respects. There is always the question of superiority and inferiority between their personalities. The word uttama is very significant. No one can surpass the Supreme Personality of Godhead.

The word loke signifies “in the paurusa agama (the smriti scriptures).” As confirmed in the Nirukti dictionary, lokyate vedartho ’nena: “The purpose of the Vedas is explained by the smriti scriptures.”

The Supreme Lord, in His localized aspect of Paramatma, is also described in the Vedas themselves.

The following verse appears in the Vedas (Chandogya Upanishad 8.12.3): tavad esa samprasado ’smac charirat samutthaya param jyoti-rupam sampadya svena rupenabhinispadyate sa uttamah purushah. “The Supersoul coming out of the body enters the impersonal brahmajyoti; then in His form He remains in His spiritual identity. That Supreme is called the Supreme Personality.” This means that the Supreme Personality is exhibiting and diffusing His spiritual effulgence, which is the ultimate illumination. That Supreme Personality also has a localized aspect as Paramatma. By incarnating Himself as the son of Satyavati and Parashara, He explains the Vedic knowledge as Vyasadeva.

Commentary by Sri Vishvanatha Chakravarthi Thakur of Gaudiya Sampradaya:

Having spoken about the object of worship of the yogis, paramatma, the Lord now speaks about the object of worship of the devotees, bhagavan.Though he in his form as Krsna is bhagavan himself, he denotes the supreme excellence of his own svarupa as Krsna using the name purusottama, he who is superior (uttama) to the paramatma (purusa). I am superior to the jiva (ksaram), and I am superior to the brahman (aksarat), and also to the unchanging person, paramatma (purusa).

yoginam api sarvesam mad-gatenantar-atmana
sraddhavan bhajate yo mam sa me yuktatamo matah

Of all yogis, he who, absorbing his mind in me, worships me with faith, is considered by me to be the best yogi.   BG 6.47

From the difference in the worshipper, one can conclude that there is a difference in the worshipable object. The word ca in the verse also indicates that Krsna is superior even to the Lord of Vaikuntha and others. This is understood from the words of Suka:

ete camsa-kalah pumsah krsnas tu bhagavan svayam

These are all parts or parts of parts. Krsna is bhagavan himself.    SB 1.3.28

Though his one sac cid ananda form is called by the names brahman,   paramatma   and   bhagavan,   actually   there   is   no difference at all between them, svarupa dvaydbhavat: there is no duality in your form. (SB 6.9.36) However, because of observing difference in the worshippers, in terms of sadhana and the results, difference has been designated.  That is to say, the sadhana respectively of the jnani, yogi and bhakta produces realization of brahman, paramatma and bhagavan. The result of jnana and yoga is mere liberation, and the result of bhakti is to become an associate of the Lord with prema. Also without bhakti, the jnanl and yogi do not even attain liberation. This is understood from the following verses:

naiskarmyam apy acyuta-bhava-varjitam na sobhate

Knowledge of self-realization, even though free from all material affinity, does not look well if devoid of a conception of the Infallible [God]. SB 1.5.12

pureha bhuman bahavo ‘pi yoginas
tvad-arpiteha nija-karma-labdhaya
vibudhya bhaktyaiva kathopanitaya
prapedire ‘njo ‘cyuta te gatim param

O almighty Lord, in the past many yogis in this world achieved the platform of devotional service by offering all their endeavors unto You and faithfully carrying out their prescribed duties. Through such devotional service, perfected by the processes of hearing and chanting about You, they came to understand You, O infallible one, and could easily surrender to You and achieve Your supreme abode. SB 10.14.5

The worshippers of brahman and paramatma must necessarily perform bhakti to bhagavan in order to achieve the final results of their sadhana. The worshippers of bhagavan however do not perform worship of brahman or paramatma to achieve the results of their sadhana.

tasman mad-bhakti-yuktasya yogino vai mad-atmanah
na jnanam na ca vairagyam prayah sreyo bhaved iha

Therefore, for a devotee engaged in My loving service, with mind fixed on Me, the cultivation of knowledge and renunciation is generally not the means of achieving the highest perfection within this world. SB 11.20.31

yat karmabhir yat tapasa jnana-vairagyatas ca yat
yogena dana-dharmena sreyobhir itarair api

sarvam mad-bhakti-yogena mad-bhakto labhate ‘njasa
svargapavargam mad-dhama kathancid yadi vanchati

Everything that can be achieved by fruitive activities, penance, knowledge, detachment, mystic yoga, charity, religious duties and all other means of perfecting life is easily achieved by My devotee through loving service unto Me. If somehow or other My devotee desires promotion to heaven, liberation, or residence in My abode, he easily achieves such benedictions. SB 1.20.31-32

ya vai sadhana-sampattih purusartha-catustaye
taya vina tad apnoti naro narayanasrayah

Even without the usual requirements for achieving the perfection of life, a person will gain that perfection if he is simply a surrendered devotee of Narayana Narayaniya

Therefore, by worship of bhagavan, it is possible to attain all of the results: material enjoyment, liberation and prema. But by the worship of brahman and paramatma alone, it is not possible to attain enjoyment, liberation or prema. Therefore, it is said that bhagavan is in a superior position even though he is not different from brahman and paramatma. It is just as the huge fire is considered superior among a small light, a lamp and a huge fire, because of its superior ability to alleviate the pain of cold, even though all the items are essential the same energy.

However, among the forms of bhagavan, Krsna is considered to be the supreme attractive, just as the sun is considered superior to the huge fire. Why? Because Krsna alone gave nirvana moksa, the final fruit of worship of brahman, even to his enemies such as Agha, Baka and Jarasandha, who were most sinful. This verse has been explained according to Sridhara Svami’s commentary on the verse brahmano hi pratisthaham.

Madhusudana Sarasvati says:

Men skilful at pious work worship Krsna repeatedly, who appears continually to relieve the burden of the earth, who enables the intelligent to cross the material ocean, who is the delight of the women of Vraja, the essence of the beauty of the rain cloud as described in the Vedas, and the form of knowledge and bliss.

I do not know any truth higher than Krsna, who has lotus eyes, a face beautiful as the full moon, and lips red like the bimba fruit; who has the complexion of a rain cloud; who wears yellow cloth; and who holds a flute in his hand. Madhusudana Sarasvati’s commentary on verse 19

Those fools who cannot tolerate the glories of Krsna which have been ascertained with valid proof go to hell. Madhusudana Sarasvati’s commentary on verse 20

There should be no indignation at the explanation of these three verses (16-18) which establish the supreme position of the form of Krsna, with all due respects to the impersonalists.

Commentary by Sri Ramanuja of Sri Sampradaya:

15.18 Inasmuch as I transcend the perishable (i.e., bound) Person of the aforesaid nature, and I am higher, for reasons stated earlier, than the imperishable Person or liberated self, therefore I am styled the Supreme Person in the Smrti and Srutis. The Smrti is called Loka by reason of its leading to the meaning of the Vedas. The meaning is that I am famous in the Srutis and in the Smrti. In the Sruti for instance; ‘Reaching the Supreme Light, it appears in its own nature. He is the Supreme Person’ (Cha. U., 8.12.3). In the Smrti we have texts like ‘I will approach Him (Sri Krsna), the Supreme Person who is the incarnation of a portion of Visnu, who is without beginning, middle or end’ (V. P., 5.17.33).

Commentary by Sri Sridhara Swami of Rudra Sampradaya:

In this verse the Supreme Lord Krishna confirms His position as Purusottama the Supreme Being of all existence by His validation of transcendence to both the ksarah or perishable and aksarah or imperishable; along with all jivas or embodied beings in bondage as well as those liberated from samsara or the perpetual cycle of birth and death. Therfore throughout creation and in the Vedic scriptures He is celebrated and reknown as Purusottama. The Brihadaranya Upanisad V.VI.I beginning manomayo-ayam puruso states: The resplendent supreme being is the ruler and lord of all governing all that exists.

Commentary by Sri Madhvacharya of Brahma Sampradaya:

Sri Madhvacharya did not comment on this sloka.

Commentary by Sri Keshava Kashmiri of Kumara Sampradaya:

One may not comprehend that the Supreme Being and Lord Krishna are the same because of the fact that Lord Krishna, Himself refers to the Supreme being as interpenetrating all the worlds. To remove any vestiges of misunderstanding Lord Krishna construes this verse with the etymological expression of the preceeding verse with Himself in the first person singular stating: Since I am transcendental to ksarah which is the perishable, the insentient, material nature as well as transcendental to aksarah the imperishable, the eternal, the atma or immortal soul; the Supreme Lord is known as Purusottama the Supreme Being. The Svetasvatara Upanisad VI.XVI beginning sa vishva krid vishva avid atma states: The Supreme Being is the creator of everything as well as the knower of everything. He is also the absolute controller of everything. He is the cause of liberation from material existence by the jivas or embodied beings and it is due to aversion to Him that jivas are entangled in samsara the perpetual cycle of birth and death. The Vishnu Sahasranama, verse 16 is states that the Supreme Lord is paramount among all beings. The Puranas glorify Him as the Supreme Sovereign of all that exists and the Narada Pancaratra states: O Supreme Lord as you are the Supreme Being you must save me for I am extremely unfortunate drowning in the ocean of material existence. In this way Lord Krishna is known in the world and in the Vedic scriptures as Purusottama.

Commentary by Sri Adi Shankaracharya of Advaita Sampradaya:

15.18 Yasmat, since; aham, I; am atitah, transcendental; ksaram, to the mutable-I am beyond the Tree of Maya, called the Peepul Tree, which this worldly existence is; and uttamah, above, most excellent or the highest; as compared with api, even; the akasarat, immutable, which is the seed of the Tree of worldly existence; atah, hence, by virtue of being the most excellent as compared with the mutable and the immutable; aham, I; am prathitah, well known; loke, in the world; and vede, in the Vedas; as purusottamah, the supreme Person. Devoted persons know Me thus, and poets also use this name ‘Purusottama’ in their poetry etc.; they extol Me with this name. Thereafter, now is stated this result attained by one who knows the Self as described:

Commentary by Sri Abhinavagupta of Kaula Tantra Sampradaya:

15.16-18 Dvav imau etc. upto Purusottamah. What has been stated in the passage ‘There are two persons in the world etc.’, is this : The body organism is made of the earth and other elements. In the world every person, unitelligent by nature, takes the body for the Self and [hence] views the Soul to be of perishing nature. Hence, the sense of duality does not come to an end with regard to the worldly persons, because of their delusion. But I am (the I-consciousness is) the One favouring all, and by cutting the daulity-knot I am to be realised as the One pervading all. (I) have transcended the perishing : Since the elements are insentient. (I) have transcended the nonperishing : Since the omnipresence [of the Self] is cut off (not comprehended) when the Self is not properly realised. In the world and in the Veda too I am acclaimed as the Highest of Persons : The Self same Supreme Self, admitting no duality, is described in this manner with the sentences ‘He is he Highest Person’ and the like.

Sanskrit Shloka Without Transliteration Marks:

yasmat ksaram atito ’ham
aksarad api cottamah
ato ’smi loke vede ca
prathitah purusottamah

Sanskrit to English Word for Word Meanings:

yasmāt — because; kṣaram — to the fallible; atītaḥ — transcendental; aham — I am; akṣarāt — beyond the infallible; api — also; ca — and; uttamaḥ — the best; ataḥ — therefore; asmi — I am; loke — in the world; vede — in the Vedic literature; ca — and; prathitaḥ — celebrated; puruṣa-uttamaḥ — as the Supreme Personality.