Over the past few years, I have provided a number of selections from Eastern Orthodox scholars/theologians concerning 'the monarchy of God the Father' and the doctrine of the Trinity. In this post I expand some of the excerpts, and add a few more.
Boris Bobrinskoy (The Mystery of the Trinity, 1999) -
The paternity of the Father is unique, ineffable, perfect, not only the mystery of the relation between the Father and the Son, but also the archetypal foundation of all human fatherhood, source of the perfect grace coming from on high, from the Father of lights (Jm 1:17): "For this reason I bow my knees before the Father, from whom every family in heaven and on earth is named" (Page 262).
Following the Cappadocians, the patristic tradition differentiates in the mystery of the Father between His "absolute," negative property of being ungenerated, and His "relative" and positive property of Paternity.
The proprium of the Hypostasis of the Father is to be "without cause," without "beginning." These negative terms carry all the weight of the Uniqueness of the Father, who is the only one not to receive His origin in the divinity from another Hypostasis. But these terms do not suffice, and the concept of "Ungenerated" specifies still more the unique character of that One who does not have origin.
"The Father is uncaused (anaitios) and ungenerated (agennētos); He is not from another, but He has being from Himself [i.e. autotheos]; and whatsoever He has, He does not have from another." 
3. St. John Damascene, De fide orthodoxa, I.8, PG 94:821D. (Page 263)
...the Father is not only "uncaused" and "ungenerated," but he is the "cause," the "principle" (archē) not only of the being of creatures, but also of the trinitarian Hypostases of the Son and of the Spirit. (Page 264)
St. Gregory Nazianzen said, "I want to call the Father greater (than the Son); this expression "greater" refers to cause, not to essence, because to those who are like essence (tōn homoousiōn) there is no greater or less in the point of essence.) 
5. Oratio XL., In sanctum baptisma, 43, PG 36:419BC. (Page 264)
Causality, then, belongs properly to the Father. This is the fundamental principle of the "monarchy". (Page 265)
The Monarchy of the Father proclaims, by necessity, the nontemporal origin of the Son and the Spirit. (Page 265)
The Father is the sole cause of the Godhead... (Page 266)
Thus, the oneness of God is placed not only on the level of the nature common to the Three, but on the basis of the personal relation or origin from the Father. (Page 266)
Vladmir Lossky (Orthodox Theology, Eng. trans. 1978, 2nd ed.) -
The term "monarch" for the Father is current in the great theologians of the fourth century. It means that the very source of divinity is personal. The Father is divinity, but precisely because he is the Father, He confers it in its fullness on the two other persons. The latter take their origin from the Father, μόνη ἄρχή, single principle, whence the term "monarchy," the divinity-source," as Dionysius the Areopagite says of the Father. It is from this indeed that springs—this that is rooted—the identical, unshared, but differently communicated divinity of the Son and the Holy Spirit. (Page 46)
John Meyendorff (Byzantine Theology, 2nd ed, 1983) -
The same personalistic emphasis appears in the Greek Fathers' insistence on the "monarchy" of the Father. Contrary to the concept which prevailed in the post-Augustinian West and in Latin Scholasticism, Greek theology attributes the origin of hypostatic "subsistence" to the hypostasis of the Father—not to the common essence. The Father is the "cause" (aitia) and the "principle" (archē) of the divine nature, which is in the Son and in the Spirit. What is even more striking is the fact that this "monarchy" of the Father is constantly used by the Cappadocian Fathers against those who accuse them of "tritheism": "God is on," writes Basil, "because the Father is one." (Page 183)
John Zizioulas (Being As Communion, 1985) -
Among the Greek Fathers the unity of God, the one God, and the ontological "principal" or "cause" of the being and life of God does not consist in the one substance of God but in the hypostasis, that is, the person of the Father. The one God is not the one substance but the Father, who is the "cause" both of the generation of the Son and the procession of the Spirit. (Pages 40, 41)
John Behr -
So how can Christians believe in and worship the Father, the Son and the Holy Spirit, and yet claim that there is only one God, not three? How can one reconcile monotheism with trinitarian faith?
My comments here follow the structure of revelation as presented in Scripture and reflected upon by the Greek Fathers of the fourth century, the age of trinitarian debates. To avoid the confusion into which explanations often fall, it is necessary to distinguish between: the one God; the one substance common to Father, Son and Holy Spirit; and the one-ness or unity of these Three.
The Father alone is the one true God. This keeps to the structure of the New Testament language about God, where with only a few exceptions, the world “God” (theos) with an article (and so being used, in Greek, as a proper noun) is only applied to the one whom Jesus calls Father, the God spoken of in the scriptures. This same fact is preserved in all ancient creeds, which begin: I believe in one God, the Father…
“For us there is one God, the Father… and one Lord Jesus Christ” (1 Cor 8:6).
The proclamation of the divinity of Jesus Christ is made no so much by describing Him as “God” (theos used, in Greek, without an article is as a predicate, and so can be used of creatures; cf. John 10:34-35), but by recognizing Him as “Lord” (Kyrios).
Beside being a common title (“sir”), this word had come to be used, in speech, for the unpronounceable, divine, name of God Hiself, YHWH. When Paul states that God bestowed upon the crucified and risen Christ the
“name above every name” (Phil 2:9),
this is an affirmation that this one is all that YHWH Himself is, without being YHWH. This is again affirmed in the creeds.
“And in one Lord Jesus Christ, the Son of God… true God of true God.”
According to the Nicene creed, the Son is
“consubstantial with the Father.”
St Athanasius, the Father who did more than anyone else to forge Nicene orthodoxy, indicated that
“what is said of the Father is said in Scripture of the Son also, all but His being called Father” (On the Synods, 49).
It is important to note how respectful such theology is of the total otherness of God in comparison with creation: such doctrines are regulative of our theological language, not a reduction of God to a being alongside other beings. It is also important to note the essential asymmetry of the relation between the Father and the Son: the Son derives from the Father; He is, as the Nicene creed put it, “of the essence of the Father” – they do not both derive from one common source. This is what is usually referred to as the Monarchy of the Father.
St Athanasius also began to apply the same argument used for defending the divinity of the Son, to a defense of the divinity of the Holy Spirit: just as the Son Himself must be fully divine if He is to save us, for only God can save, so also must Holy Spirit be divine if He is to give life to those who lie in death. Again there is an asymmetry, one which also goes back to Scripture: we receive the Spirit of Him who raised Jesus from the dead as the Spirit of Christ, one which enables us to call on God as “Abba.” Though we receive the Spirit through Christ, the Spirit proceeds only from the Father, yet this already implies the existence of the Son, and therefore that the Spirit proceeds from the Father already in relation to the Son (see especially St Gregory of Nyssa, To Ablabius: That there are not Three Gods).
So there is one God and Father, one Lord Jesus Christ, and one Holy Spirit, three “persons” (hypostases) who are the same or one in essence (ousia); three persons equally God, possessing the same natural properties, yet really God, possessing the same natural properties, yet really distinct, known by their personal characteristics. Besides being one in essence, these three persons also exist in total one-ness or unity.
There are three characteristics ways in which this unity is described by the Greek Fathers. The first is in terms of communion:
“The unity [of the three] lies in the communion of the Godhead”
as St Basil the Great puts it (On the Holy Spirit 45). The emphasis here on communion acts as a safeguard against any tendency to see the three persons as simply different manifestations of the one nature; if they were simply different modes in which the one God appears, then such an act of communion would not be possible. The similar way of expressing the divine unity is in terms of “coinherence” (perichoresis): the Father, Son and Holy Spirit indwell in one another, totally transparent and interpenetrated by the other two. This idea clearly stems from Christ’s words in the Gospel of John:
“I am in the Father and the Father in me” (14:11).
Having the Father dwelling in Him in this way, Christ reveals to us the Father, He is “the image of the invisible God” (Col 1:15).
The third way in which the total unity of the Father, Son and Holy Spirit is manifest is in their unity of work or activity. Unlike three human beings who, at best, can only cooperate, the activity of the Father, Son and Holy Spirit is one. God works, according to the image of St Irenaeus, with His two Hands, the Son and the Spirit.
“the work of God,” according to St Irenaeus, “is the fashioning of man” into the image and likeness of God (Against the Heretics 5.15.2),
a work which embraces, inseparably, both creation and salvation, for it is only realized in and by the crucified and risen One: the will of the Father is effected by the Son in the Spirit.
Such, then, is how the Greek Fathers, following Scripture, maintained that there is but one God, whose Son and Spirit are equally God, in a unity of essence and of existence, without compromising the uniqueness of the one true God. (From the online article, The Trinity: Scripture and the Greek Fathers - link - bold emphasis added)
Thomas Hopko -
... in the Bible, in the creeds, and in the Liturgy, it’s very important, really critically important, to note and to affirm and to remember that the one God in whom we believe, strictly speaking, is not the Holy Trinity. The one God is God the Father. In the Bible, the one God is the Father of Jesus Christ. He is God who sends his only-begotten Son into the world, and Jesus Christ is the Son of God. Then, of course, in a parallel manner, the Spirit, the Holy Spirit, is the Spirit of God, that the Holy Spirit, being the Spirit of God, is therefore also the Spirit of Christ, the Messiah, because the Christ is the Son of God, upon whom God the Father sends and affirms his Holy Spirit. (From the online transcript of the podcast, The Holy Trinity - link)
Grace and peace,