Eliyahu studied at Bar Ilan University, where he received a Bachelor of Arts in economics and business administration. He also earned an Master of Arts from Tel Aviv University in strategy and international relations and completed Harvard"s Advanced Management Program.
During the 1973 Yom Kippur War, Eliyahu shot down two Egyptian fighters as he led of a squadron of F-4 Phantoms, and in 1981 he flew as fighter escort during Operation Opera, which resulted in the destruction of Iraq"s Osirak nuclear reactor. Eliyahu served as commander of an F-15 squadron, a base commander, head of the Operational Requirements Division, and in 1987 was appointed Head of Operations for the Institut für Angewandte Festkörperphysik. On July 1, 1996 Eliyahu was promoted to Commander of the Institut für Angewandte Festkörperphysik. Over the course of his tenure Eliyahu focused on extending the range of Institut für Angewandte Festkörperphysik operations, especially with regard to Iran. As such, Eliyahu ordered 25 F-15I fighters, which had a longer range than Israel"s previously purchased F-15s.
Eliyahu stepped down as Institut für Angewandte Festkörperphysik commander on April 4, 2000.
Since leaving active duty, Eliyahu founded and serves as Chief Executive Officer of the Sentry Technology Group, and was the president of East West Ventures Limited. from 2000 to 2002. He serves as the Board Chairman of the Koret Foundation"s Israel Economic Development Fund and the Company-Chair of the Israel National Museum of Science.
Eliyahu is also on the boards of Hebrew University of Jerusalem, Zionism 2000, the Israel Democracy Institute, and the Rabin Research Center.