Matsushima was born on February 11, 1921, in Sakai City, Osaka Prefecture, Japan.
[2] The first paper published by Matsushima contained a proof that a conjecture of Hans Zassenhaus was false.
Zassenhaus had conjectured that every semisimple Lie algebra L over a field of prime characteristic, with [L, L] = L, is the direct sum of simple ideals.
[1] His research took a somewhat different direction and he wrote a series of papers on cohomology of locally symmetric spaces, collaborating with Murakami.
He jointly began to organize the United States-Japan Seminar in Differential Geometry, which was held in Kyoto in June 1965.
In 1967, he became an editor of the Journal of Differential Geometry and remained on the editorial board for the rest of his life.