A morphogenic matrix is the cellular and quantum structure of a changeling, which allows them to assume various shapes or forms. It also allows Changelings to shunt some of their mass into another dimension, thereby allowing them to morph into objects with a much smaller mass than themselves.
Although capable of shape-shifting in almost any temperature, a changeling's morphogenic matrix is most malleable at temperatures of seventeen degrees Celsius. This determination was arrived at by Dr. Mora Pol after weeks of studying the shape-shifting abilities of Odo. (DS9: "The Begotten")
A changeling's morphogenic matrix could be destabilized through exposure to polaron radiation, forcing a changeling to revert to his natural, gelatinous state. In 2373, Federation officers from Deep Space 9 attempted to use polaron radiation on a changeling whom they believed was impersonating Klingon Chancellor Gowron. (DS9: "Apocalypse Rising")
When a baby changeling was discovered later that year, Dr. Julian Bashir determined that the baby changeling had been subjected to tetryon radiation. After purging the radioactive isotopes, Bashir noted that the baby's morphogenic matrix still had a small degree of instability. The baby would ultimately die when its morphogenic matrix destabilized. (DS9: "The Begotten")
Upon encountering another Changling named Laas in 2375, a medical scan by Bashir revealed that Laas' morphogenic matrix had not been infected with the morphogenic virus which had infected the rest of the Great Link. (DS9: "Chimera")
In 2375, Bashir "borrowed" a sample of Odo in order to study the changeling's morphogenic matrix in an attempt to synthesize organic tissue with the ability to change form. It was during the study of the matrix that Bashir learned that Odo had been infected with the same morphogenic disease that was affecting his people. (DS9: "When It Rains...")