Iaijutsu is a traditional Japanese martial art which has existed largely unchanged for 500 years. Whilst technically there are differences between Iai, Iaijutsu and Iaido, generally it's accurate to think of these all as the art of drawing and cutting with the Japanese sword. Iaijutsu techniques usually consist of a simultaneous draw and cut, often a follow up attack, cleaning of the blade and finally sheathing of the blade. These techniques are performed very slowly compared to other martial arts, with a focus on technical correctness and timing. Iaijutsu embodies the spirit of Japanese Budo and culture, and represents an authentic and fulfilling path on the Martial Way, either on it's own or as a compliment to other traditional budo styles.
If you would like to see some traditional Iaijutsu techniques the below link will take you to a search result on youtube, please note that these are practitioners from around the world, not specifically the Jikyo-juku school.
Youtube Search Results for MJER Iaijustu "Mae"
The Muso Jikiden Eishin Ryu entry on Wikipedia is a good starting point for those wanting to learn about the history of Iaijutsu.
Muso Jikiden Eishin Ryu on Wikipedia