Transform theory

In mathematics, transform theory is the study of transforms, which relate a function in one domain to another function in a second domain. The essence of transform theory is that by a suitable choice of basis for a vector space a problem may be simplified—or "diagonalized" as in spectral theory.
Spectral theory.
In spectral theory, the spectral theorem says that if "A" is an "n"×"n" self-adjoint matrix, there is an orthonormal basis of eigenvectors of "A". This implies that "A" is diagonalizable.
Furthermore, each eigenvalue is real.