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Here is the comment of Ryrie in the “Ryrie Study Bible.”
"The genealogy in Matt. 1:1f is traced through Joseph, Jesus’ legal (though not natural) father, and it establishes His claim and right to the throne of David (v. 6). The genealogy in Luke 3:23-38 is evidently that of Mary, though some believe it is also Joseph’s, by assuming that Matthan (Matt 1:15) and Matthat (Luke 3:24) were the same person and Jacob (Matt 1:16) and Eli (Luke 3:23) were brothers (one being Joseph’s father and the other his uncle). See note on Luke 3:23.
Many conservative scholars have taken this view because of the many differences in the names between the two accounts. The biggest difference is that after David, many of the names are different. It would seem best to explain the difference by viewing Luke’s genealogy as tracing the physical ancestors of Christ through Mary, while Matthew’s genealogy traces the kingly line of Christ through Joseph.
The following is the comment from The Bible Knowledge Commentary:
In addition Luke’s and Matthew’s lists from David to Shealtiel (during the time of the Exile) differ. That is because the lists trace different lines. Luke traced David’s line through Nathan, whereas Matthew traced it through Solomon. Following Shealtiel’s son, Zerubbabel, the lists once again differ until both lists unite at Joseph whom, Luke noted, was “thought” to be the father of Jesus. Little doubt exists that Matthew’s genealogy traced the kingly line of David—the royal legal line. The question is, What is the significance of Luke’s genealogy? Two main possibilities exist.
1. Luke was tracing the line of Mary. Many interpreters argue that Luke was giving the genealogy of Mary, showing that she also was in the line of David and that therefore Jesus was qualified as the Messiah not only through Joseph (since he was the oldest legal heir) but also through Mary.
2. Luke was tracing the actual line of Joseph. This view maintains that the legal line and the actual line of David through which Jesus came met at Joseph, the supposed father of Jesus. In this view Jacob, Joseph’s uncle, would have died childless and therefore Joseph would have been the closest living heir. Thus Joseph and then Jesus would have been brought into the royal line.
Both views have problems which are difficult to answer, not the least of which is the fact that the two genealogies meet at Shealtiel and Zerubbabel and then split a second time only to come together at Joseph and Jesus. (Cf. comments on Matt. 1:12.) Regardless of one’s view it is important to note an important aspect of the theology Luke expressed in his genealogy. He related Jesus not only to Abraham but all the way back to Adam and to God. This is an indication of the universal offer of salvation, which is common to his Gospel—that Jesus came to save all people—Gentiles as well as the nation of Israel (cf. Luke 2:32).
Luke 3:23 says, “And when He began His ministry, Jesus Himself was about thirty years of age, being supposedly the son of Joseph, the son of Eli.” Many believe that Luke is saying that Jesus was the grandson of Eli or Heli through Mary. Eli was Mary’s father and Jesus’ grandfather. By contrast, Joseph was son of Jacob according to Matthew.
There is another argument that comes from the theme, purpose, and audience of the two gospels. Matthew was written to the Jews to prove that Jesus was in the legal line of David by adoption through Joseph. However, this was not Luke’s purpose. Luke was writing to show and emphasize the humanity of Christ. He was writing to Gentiles or Greeks to show Jesus’ involvement with the needs of men. In keeping with this focus, we might naturally expect Luke, the doctor, to present the genealogy of Jesus through Mary, the source of his true humanity. For further study you might check out the studies on “Prophecies of the Birth of Christ” on of our web site."
INSIDE EVERY PROGRESSIVE IS A TOTALITARIAN, SCREAMING TO GET OUT. (David Horowitz)